Introduction
A basic problem in additive combinatorics is to determine which finite patterns must occur in dense subsets of integer lattices. At the qualitative level, the multidimensional Szemerédi theorem of Furstenberg and Katznelson [7 ] gives a general answer: for every finite set F ⊆ Z d F \subseteq\mathbb{Z}^d F ⊆ Z d Z d \mathbb{Z}^d Z d x + q F x+qF x + q F x ∈ Z d x \in\mathbb{Z}^d x ∈ Z d q ∈ N q \in\mathbb{N} q ∈ N [9 , 28 ] .
Here we study the corresponding quantitative problem for similarity classes of simplices. Let
Δ = { v 0 , … , v K − 1 } ⊆ Z d \Delta= \{v_0,\ldots,v_{K-1}\} \subseteq\mathbb{Z}^d Δ = { v 0 , … , v K − 1 } ⊆ Z d be the vertex set of a fixed nondegenerate ( K − 1 ) (K-1) ( K − 1 ) Δ ′ = { x 0 , … , x K − 1 } ⊆ Z d \Delta' = \{x_0,\ldots,x_{K-1}\} \subseteq\mathbb{Z}^d Δ ′ = { x 0 , … , x K − 1 } ⊆ Z d similar copy of Δ \Delta Δ
∣ x i − x j ∣ 2 = t ∣ v i − v j ∣ 2 ( 0 ≤ i < j ≤ K − 1 ) (1) \lvert x_i-x_j\rvert^2=t\lvert v_i-v_j\rvert^2 \qquad(0 \leq i < j \leq K-1) \tag*{(1)} ∣ x i − x j ∣ 2 = t ∣ v i − v j ∣ 2 ( 0 ≤ i < j ≤ K − 1 ) (1) for some t > 0 t>0 t > 0 Δ ′ \Delta' Δ ′ Δ \Delta Δ
Our main result is the following.
Theorem 1.1. Let K ≥ 3 K \geq3 K ≥ 3 Δ ⊆ Z d \Delta\subseteq\mathbb{Z}^d Δ ⊆ Z d ( K − 1 ) (K-1) ( K − 1 ) d ≥ 4 K + 4 d \geq4K+4 d ≥ 4 K + 4 C , c > 0 C,c>0 C , c > 0 Δ \Delta Δ d d d A ⊆ [ N ] d A \subseteq[N]^d A ⊆ [ N ] d Δ \Delta Δ
∣ A ∣ ≤ C N d exp ( − c log N ) . \lvert A\rvert\leq CN^d\exp\left(-c\sqrt{\log N}\right). ∣ A ∣ ≤ C N d exp ( − c log N ) . In the common range d ≥ 4 K + 4 d \geq4K+4 d ≥ 4 K + 4 [22 ] , which states that, when d ≥ 2 K + 3 d \geq2K+3 d ≥ 2 K + 3 A ⊆ [ N ] d A \subseteq[N]^d A ⊆ [ N ] d Δ \Delta Δ
∣ A ∣ ≤ C Δ , d N d ( log N ) − 2 / ( 11 K ) . \lvert A\rvert\leq C_{\Delta,d}N^d(\log N)^{-2/(11K)}. ∣ A ∣ ≤ C Δ , d N d ( log N ) − 2/ ( 11 K ) . To the best of our knowledge, this is the only previous quantitative result formulated for similar copies of an arbitrary nondegenerate integral simplex.
By Chebyshev’s bound, Theorem 1.1 immediately gives the following quantitative relative result in the prime lattice; compare this to [4 , 6 , 29 ] . Let P \mathbb{P} P
Corollary 1.2. Let K ≥ 3 K \geq3 K ≥ 3 Δ ⊆ Z d \Delta\subseteq\mathbb{Z}^d Δ ⊆ Z d ( K − 1 ) (K-1) ( K − 1 ) d ≥ 4 K + 4 d \geq4K+4 d ≥ 4 K + 4 C , c > 0 C,c>0 C , c > 0 Δ \Delta Δ d d d A ⊆ ( P ∩ [ N ] ) d A \subseteq(\mathbb{P} \cap[N])^d A ⊆ ( P ∩ [ N ] ) d Δ \Delta Δ
∣ A ∣ ≤ C ∣ P ∩ [ N ] ∣ d exp ( − c log N ) . \lvert A\rvert\leq C\lvert\mathbb{P}\cap[N]\rvert^d\exp\left(-c\sqrt{\log N}\right). ∣ A ∣ ≤ C ∣ P ∩ [ N ] ∣ d exp ( − c log N ) . An equivalent form of our main result is that a set A ⊆ [ N ] d A \subseteq[N]^d A ⊆ [ N ] d α \alpha α
N ≥ exp ( C Δ , d ( 1 + log ( 1 / α ) ) 2 ) . (2) N \geq\exp\left(C_{\Delta,d}\left(1+\log(1/\alpha)\right)^2\right). \tag*{(2)} N ≥ exp ( C Δ , d ( 1 + log ( 1/ α ) ) 2 ) . (2) Theorem 1.1 also directly implies the following quantitative result in the Euclidean setting.
Corollary 1.3. Let K ≥ 3 K \ge3 K ≥ 3 d ≥ 4 K + 4 d \ge4K+4 d ≥ 4 K + 4 Δ ⊆ Z d \Delta\subseteq\mathbb{Z}^{d} Δ ⊆ Z d E ⊆ [ 0 , 1 ] d E \subseteq[0,1]^{d} E ⊆ [ 0 , 1 ] d ∣ E ∣ ≥ α > 0 |E| \ge\alpha> 0 ∣ E ∣ ≥ α > 0 x ∈ R d x \in\mathbb{R}^{d} x ∈ R d U ∈ O ( d ) U \in\operatorname{O}(d) U ∈ O ( d )
λ ≥ exp ( − C Δ , d ( 1 + log ( 1 / α ) ) 2 ) \lambda\ge\exp\left(-C_{\Delta,d}(1+\log(1/\alpha))^{2}\right) λ ≥ exp ( − C Δ , d ( 1 + log ( 1/ α ) ) 2 ) such that x + λ U Δ ⊆ E x+\lambda U\Delta\subseteq E x + λ U Δ ⊆ E
A short proof, based on averaging Theorem 1.1 over translates of a sufficiently fine lattice, is given in Appendix A. For comparison, a polylogarithmic bound follows from the proof of Bourgain’s simplex theorem in dimensions d ≥ K d \ge K d ≥ K [1 , 21 ] . The cut-operator method also has a direct and simpler implementation in the Euclidean setting, which should permit a lower dimension threshold; we do not pursue this direction here.
Beyond [22 ] , quantitative results in the literature concern special configurations or indirect parametrizations by nonsingular matrix progressions. A notable example is the corners configuration, consisting of axis-parallel isosceles right triangles. Building on ideas originating in the work of Kelley and Meka [15 ] , Jaber, Liu, Lovett, Ostuni and Sawhney [14 ] recently proved that every corner-free set A ⊆ [ N ] 2 A \subseteq[N]^{2} A ⊆ [ N ] 2
∣ A ∣ ≤ N 2 exp ( − c ( log N ) 1 / 600 ) . |A| \le N^{2}\exp\left(-c(\log N)^{1/600}\right). ∣ A ∣ ≤ N 2 exp ( − c ( log N ) 1/600 ) . For an integral planar triangle, complex multiplication gives a parametrization of its similar copies in the form x x x x + T 1 y x+T_{1}y x + T 1 y x + T 2 y x+T_{2}y x + T 2 y T 1 T_{1} T 1 T 2 T_{2} T 2 T 1 − T 2 T_{1}-T_{2} T 1 − T 2 [24 ] gives a density bound of the form ( log N ) − 1 − c (\log N)^{-1-c} ( log N ) − 1 − c c > 0 c>0 c > 0 [8 ] , similarly give four-term parametrizations of tetrahedral shapes after passage to a suitable auxiliary lattice. Prendiville’s theorem [25 ] then yields a density bound of the form ( log log N ) − c (\log\log N)^{-c} ( log log N ) − c
This Hurwitz–Radon construction extends to arbitrary nondegenerate simplices, and more generally to configurations of fixed affine rank, but it has an unavoidable dimensional cost. The sharp Hurwitz–Radon theorem forces a ( K − 1 ) (K-1) ( K − 1 ) exp ( Ω ( K ) ) \exp(\Omega(K)) exp ( Ω ( K )) [8 ] show that the optimal families may be chosen integral. Prendiville’s work also suggests that longer nonsingular matrix progressions should be accessible through higher-order Fourier analysis. Together with the recent quasipolynomial inverse theorem of Leng, Sah and Sawhney [19 , 18 ] , this suggests that one might hope for a density bound of the form
exp ( − ( log log N ) c K ) \exp\left(-(\log\log N)^{cK}\right) exp ( − ( log log N ) cK ) for each fixed nonsingular matrix progression. Such a result is not presently available: a multidimensional localized density-increment argument would still have to be developed. Even if carried out, this route would give a substantially weaker dependence on N N N Theorem 1.1 and, through the Hurwitz–Radon parametrization, would require an exponentially large dimension.
Acknowledgments and AI disclosure
This project was initiated while the authors were supported by the HUN-REN Alfréd Rényi Institute of Mathematics (Erdős Center) from January to June 2026. The authors thank the Institute for its hospitality.
The second author was supported by Simons Foundation grant MPS-TSM-854813 and grant NFKIH Excellence 15421 from the Research Development and Innovation Office of Hungary.
Large language models were used as a sounding board during the brainstorming phase of the project. They were also used for literature search and copyediting. However, the final argument is the authors’ own, and the manuscript was entirely written and verified by the authors, who take full responsibility for the contents of the proof.
Outline of the proof
Setup
Let F 0 , … , F K − 1 : [ N ] d → [ − 1 , 1 ] F_0,\ldots,F_{K-1}:[N]^d\to[-1,1] F 0 , … , F K − 1 : [ N ] d → [ − 1 , 1 ] Z d \mathbb{Z}^d Z d L : = ( K 2 ) L:=\binom{K}{2} L := ( 2 K ) w = 1 [ 1 , 2 ) w=\mathbf{1}_{[1,2)} w = 1 [ 1 , 2 ) R = ⌊ σ N ⌋ R=\lfloor\sigma N\rfloor R = ⌊ σ N ⌋ 0 < σ < 1 0<\sigma<1 0 < σ < 1 Δ \Delta Δ N N N Δ \Delta Δ d d d w R ( n ) = R − 2 w ( n / R 2 ) w_R(n)=R^{-2}w(n/R^2) w R ( n ) = R − 2 w ( n / R 2 ) (1) for t = n ∈ N t=n\in\mathbb{N} t = n ∈ N n ≍ R 2 n\asymp R^2 n ≍ R 2
Λ N ( F 0 , … , F K − 1 ) = ∑ n w R ( n ) ∑ z 0 , … , z K − 1 ∈ [ N ] d ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( i < j ) ∏ i = 0 K − 1 F i ( z i ) . (3) \Lambda_N(F_0,\ldots,F_{K-1})=\sum_n w_R(n)\sum_{\substack{z_0,\ldots,z_{K-1}\in[N]^d\\ |z_i-z_j|^2=n|v_i-v_j|^2\ (i<j)}}\prod_{i=0}^{K-1}F_i(z_i). \tag*{(3)} Λ N ( F 0 , … , F K − 1 ) = n ∑ w R ( n ) z 0 , … , z K − 1 ∈ [ N ] d ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( i < j ) ∑ i = 0 ∏ K − 1 F i ( z i ) . (3) Kitaoka’s estimate [17 ] supplies
Λ N ( 1 [ N ] d , … , 1 [ N ] d ) ≳ N d R ( K − 1 ) d − 2 L . (4) \Lambda_N(\mathbf{1}_{[N]^d},\ldots,\mathbf{1}_{[N]^d})\gtrsim N^dR^{(K-1)d-2L}. \tag*{(4)} Λ N ( 1 [ N ] d , … , 1 [ N ] d ) ≳ N d R ( K − 1 ) d − 2 L . (4) Here ( K − 1 ) d (K-1)d ( K − 1 ) d 2 L = 2 ( K 2 ) 2L=2\binom{K}{2} 2 L = 2 ( 2 K ) ( K 2 ) \binom{K}{2} ( 2 K )
Suppose that A ⊂ [ N ] d A\subset[N]^d A ⊂ [ N ] d α > 0 \alpha>0 α > 0 Δ \Delta Δ Λ N ( 1 A , … , 1 A ) = 0 \Lambda_N(\mathbf{1}_A,\ldots,\mathbf{1}_A)=0 Λ N ( 1 A , … , 1 A ) = 0 1 A = α 1 [ N ] d + g \mathbf{1}_A=\alpha\mathbf{1}_{[N]^d}+g 1 A = α 1 [ N ] d + g F 0 , … , F K − 1 F_0,\ldots,F_{K-1} F 0 , … , F K − 1
∣ Λ N ( F 0 , … , F K − 1 ) ∣ ≳ α s N d R ( K − 1 ) d − 2 L , (5) |\Lambda_N(F_0,\ldots,F_{K-1})|\gtrsim\alpha^sN^dR^{(K-1)d-2L}, \tag*{(5)} ∣ Λ N ( F 0 , … , F K − 1 ) ∣ ≳ α s N d R ( K − 1 ) d − 2 L , (5) with exactly s > 0 s>0 s > 0 F i F_i F i g g g F i F_i F i 1 [ N ] d \mathbf{1}_{[N]^d} 1 [ N ] d
As is customary in the circle method, one expresses the quadratic relations via the integrals
1 ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 = ∫ T e ( β i j ∣ z i − z j ∣ 2 ) e ( − n β i j ∣ v i − v j ∣ 2 ) d β i j , e ( t ) : = exp ( 2 π i t ) . \mathbf{1}_{|z_i-z_j|^2=n|v_i-v_j|^2}=\int_{\mathbb{T}}e(\beta_{ij}|z_i-z_j|^2)e(-n\beta_{ij}|v_i-v_j|^2)\,d\beta_{ij},\qquad e(t):=\exp(2\pi\mathrm{i}t). 1 ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 = ∫ T e ( β ij ∣ z i − z j ∣ 2 ) e ( − n β ij ∣ v i − v j ∣ 2 ) d β ij , e ( t ) := exp ( 2 π i t ) . This gives
Λ N ( F 0 , … , F K − 1 ) = ∫ T L w R ^ ( ∑ i < j ∣ v i − v j ∣ 2 β i j ) S β ( F 0 , … , F K − 1 ) d β , ∥ w R ^ ∥ ∞ ≲ 1 , (6) \Lambda_N(F_0,\ldots,F_{K-1})=\int_{\mathbb{T}^L}\widehat{w_R}\left(\sum_{i<j}|v_i-v_j|^2\beta_{ij}\right)S_\beta(F_0,\ldots,F_{K-1})\,d\beta,\qquad\|\widehat{w_R}\|_\infty\lesssim1, \tag*{(6)} Λ N ( F 0 , … , F K − 1 ) = ∫ T L w R ( i < j ∑ ∣ v i − v j ∣ 2 β ij ) S β ( F 0 , … , F K − 1 ) d β , ∥ w R ∥ ∞ ≲ 1 , (6) where w R ^ ( t ) : = ∑ n w R ( n ) e ( − n t ) \widehat{w_R}(t):=\sum_nw_R(n)e(-nt) w R ( t ) := ∑ n w R ( n ) e ( − n t )
S β ( F 0 , … , F K − 1 ) = ∑ z 0 , … , z K − 1 ∈ [ N ] d e ( ∑ i < j β i j ∣ z i − z j ∣ 2 ) ∏ i = 0 K − 1 F i ( z i ) . (7) S_\beta(F_0,\ldots,F_{K-1})=\sum_{z_0,\ldots,z_{K-1}\in[N]^d}e\left(\sum_{i<j}\beta_{ij}|z_i-z_j|^2\right)\prod_{i=0}^{K-1}F_i(z_i). \tag*{(7)} S β ( F 0 , … , F K − 1 ) = z 0 , … , z K − 1 ∈ [ N ] d ∑ e ( i < j ∑ β ij ∣ z i − z j ∣ 2 ) i = 0 ∏ K − 1 F i ( z i ) . (7) The cut-operator reduction
Write E \mathcal{E} E K K K L = ∣ E ∣ = ( K 2 ) L = |\mathcal{E}| = \binom{K}{2} L = ∣ E ∣ = ( 2 K ) β j i = β i j \beta_{ji} = \beta_{ij} β j i = β ij i < j i < j i < j
Fix a balanced cut S ∣ T S \mid T S ∣ T Q β ( z ) Q_{\beta}(z) Q β ( z ) (7) decomposes as
Q β ( z ) = Q β S ( z S ) + Q β T ( z T ) − 2 ∑ i ∈ S , j ∈ T β i j z i ⋅ z j . Q_{\beta}(z) = Q_{\beta}^{S}(z_{S}) + Q_{\beta}^{T}(z_{T}) - 2\sum_{i\in S,j\in T}\beta_{ij}z_i\cdot z_j. Q β ( z ) = Q β S ( z S ) + Q β T ( z T ) − 2 i ∈ S , j ∈ T ∑ β ij z i ⋅ z j . where z S = ( z i ) i ∈ S z_{S} = (z_i)_{i\in S} z S = ( z i ) i ∈ S z T = ( z j ) j ∈ T z_{T} = (z_j)_{j\in T} z T = ( z j ) j ∈ T Θ ∈ T ∣ S ∣ ∣ T ∣ \Theta\in\mathbb{T}^{|S||T|} Θ ∈ T ∣ S ∣∣ T ∣
K R , d S , T ( Θ ) : ℓ 2 ( ( [ R ] d ) T ) → ℓ 2 ( ( [ R ] d ) S ) \mathcal{K}_{R,d}^{S,T}(\Theta):\ell^{2}(([R]^d)^T)\to\ell^{2}(([R]^d)^S) K R , d S , T ( Θ ) : ℓ 2 (([ R ] d ) T ) → ℓ 2 (([ R ] d ) S ) defined by
( K R , d S , T ( Θ ) h ) ( z S ) : = ∑ z T ∈ ( [ R ] d ) T e ( ∑ i ∈ S , j ∈ T Θ i j z i ⋅ z j ) h ( z T ) . (\mathcal{K}_{R,d}^{S,T}(\Theta)h)(z_S):=\sum_{z_T\in([R]^d)^T}e\left(\sum_{i\in S,j\in T}\Theta_{ij}z_i\cdot z_j\right)h(z_T). ( K R , d S , T ( Θ ) h ) ( z S ) := z T ∈ ([ R ] d ) T ∑ e i ∈ S , j ∈ T ∑ Θ ij z i ⋅ z j h ( z T ) . We call K R , d S , T ( Θ ) \mathcal{K}_{R,d}^{S,T}(\Theta) K R , d S , T ( Θ ) cut operator . Because the dot product separates coordinatewise, K R , d S , T ( Θ ) \mathcal{K}_{R,d}^{S,T}(\Theta) K R , d S , T ( Θ ) d d d K R S , T ( Θ ) : = K R , 1 S , T ( Θ ) \mathcal{K}_{R}^{S,T}(\Theta):=\mathcal{K}_{R,1}^{S,T}(\Theta) K R S , T ( Θ ) := K R , 1 S , T ( Θ )
m R S , T ( Θ ) : = R − K / 2 ∥ K R S , T ( Θ ) ∥ 2 → 2 , ( 0 ≤ m R S , T ( Θ ) ≤ 1 ) . (8) m_{R}^{S,T}(\Theta):=R^{-K/2}\|\mathcal{K}_{R}^{S,T}(\Theta)\|_{2\to2},\qquad(0\le m_{R}^{S,T}(\Theta)\le1). \tag*{(8)} m R S , T ( Θ ) := R − K /2 ∥ K R S , T ( Θ ) ∥ 2 → 2 , ( 0 ≤ m R S , T ( Θ ) ≤ 1 ) . (8) Let β S , T = ( β i j ) i ∈ S , j ∈ T \beta_{S,T}=(\beta_{ij})_{i\in S,j\in T} β S , T = ( β ij ) i ∈ S , j ∈ T S ∣ T S\mid T S ∣ T
S β ( F 0 , … , F K − 1 ) = ⟨ U S K N , d S , T ( − 2 β S , T ) U T H T , H S ⟩ , \mathcal{S}_{\beta}(F_0,\ldots,F_{K-1})=\langle U_S\mathcal{K}_{N,d}^{S,T}(-2\beta_{S,T})U_TH_T,H_S\rangle, S β ( F 0 , … , F K − 1 ) = ⟨ U S K N , d S , T ( − 2 β S , T ) U T H T , H S ⟩ , where
H S ( z S ) : = ∏ i ∈ S F i ( z i ) , H T ( z T ) : = ∏ j ∈ T F j ( z j ) , H_S(z_S):=\prod_{i\in S}F_i(z_i),\qquad H_T(z_T):=\prod_{j\in T}F_j(z_j), H S ( z S ) := i ∈ S ∏ F i ( z i ) , H T ( z T ) := j ∈ T ∏ F j ( z j ) , and U S , U T U_S,U_T U S , U T
( U S h ) ( z S ) : = e ( Q β S ( z S ) ) h ( z S ) , ( U T h ) ( z T ) : = e ( Q β T ( z T ) ) h ( z T ) . (U_Sh)(z_S):=e(Q_{\beta}^{S}(z_S))h(z_S),\qquad(U_Th)(z_T):=e(Q_{\beta}^{T}(z_T))h(z_T). ( U S h ) ( z S ) := e ( Q β S ( z S )) h ( z S ) , ( U T h ) ( z T ) := e ( Q β T ( z T )) h ( z T ) . Since ∥ H S ∥ 2 = ∏ i ∈ S ∥ F i ∥ 2 \|H_S\|_2=\prod_{i\in S}\|F_i\|_2 ∥ H S ∥ 2 = ∏ i ∈ S ∥ F i ∥ 2 ∥ H T ∥ 2 = ∏ j ∈ T ∥ F j ∥ 2 \|H_T\|_2=\prod_{j\in T}\|F_j\|_2 ∥ H T ∥ 2 = ∏ j ∈ T ∥ F j ∥ 2 ℓ 2 → ℓ 2 \ell^2\to\ell^2 ℓ 2 → ℓ 2
∣ S β ( F 0 , … , F K − 1 ) ∣ ≤ N K d / 2 m N S , T ( − 2 β S , T ) d ∏ i = 0 K − 1 ∥ F i ∥ 2 . (9) |\mathcal{S}_{\beta}(F_0,\ldots,F_{K-1})|\le N^{Kd/2}m_N^{S,T}(-2\beta_{S,T})^d\prod_{i=0}^{K-1}\|F_i\|_2. \tag*{(9)} ∣ S β ( F 0 , … , F K − 1 ) ∣ ≤ N K d /2 m N S , T ( − 2 β S , T ) d i = 0 ∏ K − 1 ∥ F i ∥ 2 . (9) The problem of estimating the exponential sum S β \mathcal{S}_{\beta} S β [11 , 12 ] , Liu [20 ] , Cook-Magyar [3 ] , and Green [10 ] .
We consider all balanced cuts of the vertex set Δ \Delta Δ B K \mathcal{B}_K B K
w K : = L ∣ S ∣ ∣ T ∣ ∣ B K ∣ , τ : = d ∣ S ∣ ∣ T ∣ 2 L . w_K:=\frac{L}{|S||T||\mathcal{B}_K|},\qquad\tau:=\frac{d|S||T|}{2L}. w K := ∣ S ∣∣ T ∣∣ B K ∣ L , τ := 2 L d ∣ S ∣∣ T ∣ . By symmetry it is easy to see that
w K # { S ∣ T ∈ B K : e crosses S ∣ T } = 1 ( e ∈ E ) . (10) w_K\#\{S\mid T\in\mathcal{B}_K:e\text{ crosses }S\mid T\}=1\qquad(e\in\mathcal{E}). \tag*{(10)} w K # { S ∣ T ∈ B K : e crosses S ∣ T } = 1 ( e ∈ E ) . (10) Since 2 τ w K = d / ∣ B K ∣ 2\tau w_K=d/|\mathcal{B}_K| 2 τ w K = d /∣ B K ∣ (9) over S ∣ T ∈ B K S\mid T\in\mathcal{B}_K S ∣ T ∈ B K
∣ S β ( F 0 , … , F K − 1 ) ∣ ≤ N K d / 2 ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K ∏ i = 0 K − 1 ∥ F i ∥ 2 . (11) |\mathcal{S}_{\beta}(F_0,\ldots,F_{K-1})|\le N^{Kd/2}\prod_{S\mid T\in\mathcal{B}_K}m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K}\prod_{i=0}^{K-1}\|F_i\|_2. \tag*{(11)} ∣ S β ( F 0 , … , F K − 1 ) ∣ ≤ N K d /2 S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K i = 0 ∏ K − 1 ∥ F i ∥ 2 . (11) The mean-value estimate
The first principal estimate is a high-moment bound for the cut operator norm.
Lemma 3.1. Let S ∣ T S\mid T S ∣ T R R R
∫ T ∣ S ∣ ∣ T ∣ m R S , T ( Θ ) 2 K + 4 d Θ ≪ K R − 2 ∣ S ∣ ∣ T ∣ . (12) \int_{\mathbb{T}^{|S||T|}}m_R^{S,T}(\Theta)^{2K+4}\,\mathrm{d}\Theta\ll_K R^{-2|S||T|}. \tag*{(12)} ∫ T ∣ S ∣∣ T ∣ m R S , T ( Θ ) 2 K + 4 d Θ ≪ K R − 2∣ S ∣∣ T ∣ . (12) The power R − 2 ∣ S ∣ ∣ T ∣ R^{-2|S||T|} R − 2∣ S ∣∣ T ∣ ∣ S ∣ ∣ T ∣ |S||T| ∣ S ∣∣ T ∣
∥ K R S , T ( Θ ) ∥ 2 → 2 2 K + 4 = ∥ ( K R S , T ( Θ ) K R S , T ( Θ ) ∗ ) K + 2 ∥ 2 → 2 \|\mathcal{K}_R^{S,T}(\Theta)\|_{2\to2}^{2K+4}=\|(\mathcal{K}_R^{S,T}(\Theta)\mathcal{K}_R^{S,T}(\Theta)^*)^{K+2}\|_{2\to2} ∥ K R S , T ( Θ ) ∥ 2 → 2 2 K + 4 = ∥ ( K R S , T ( Θ ) K R S , T ( Θ ) ∗ ) K + 2 ∥ 2 → 2 and that the latter norm is bounded by the trace of ( K R S , T ( Θ ) K R S , T ( Θ ) ∗ ) K + 2 (\mathcal{K}_R^{S,T}(\Theta)\mathcal{K}_R^{S,T}(\Theta)^*)^{K+2} ( K R S , T ( Θ ) K R S , T ( Θ ) ∗ ) K + 2 Θ \Theta Θ R K R^K R K
U ∈ Z ∣ S ∣ × ( K + 1 ) , V ∈ Z ∣ T ∣ × ( K + 1 ) , U V T = 0. (13) U\in\mathbb{Z}^{|S|\times(K+1)},\qquad V\in\mathbb{Z}^{|T|\times(K+1)},\qquad UV^{\mathsf T}=0. \tag*{(13)} U ∈ Z ∣ S ∣ × ( K + 1 ) , V ∈ Z ∣ T ∣ × ( K + 1 ) , U V T = 0. (13) We follow a geometric route exploiting the special structure of this matrix equation. In particular, after treating U = 0 U=0 U = 0 V = 0 V=0 V = 0 Γ U \Gamma_U Γ U Γ V \Gamma_V Γ V U U U V V V Γ U ⊆ Γ V ⊥ ⊆ Z K + 1 \Gamma_U\subseteq\Gamma_V^\perp\subseteq\mathbb{Z}^{K+1} Γ U ⊆ Γ V ⊥ ⊆ Z K + 1 Γ V ⊥ \Gamma_V^\perp Γ V ⊥ Γ V \Gamma_V Γ V [16 ] , gives the upper bound
# { ( U , V ) : U V T = 0 , ∥ U ∥ ∞ , ∥ V ∥ ∞ ≪ R } ≪ K R K ( K + 1 ) − 2 ∣ S ∣ ∣ T ∣ . \#\{(U,V):UV^{\mathsf T}=0,\ \|U\|_\infty,\|V\|_\infty\ll R\}\ll_K R^{K(K+1)-2|S||T|}. # {( U , V ) : U V T = 0 , ∥ U ∥ ∞ , ∥ V ∥ ∞ ≪ R } ≪ K R K ( K + 1 ) − 2∣ S ∣∣ T ∣ . Combining this with the R K R^K R K R − K ( K + 2 ) R^{-K(K+2)} R − K ( K + 2 ) [5 ] turns the one-cut estimates into the global estimate used in both the major and minor arc estimates.
Corollary 3.2. If τ ≥ K + 2 \tau\ge K+2 τ ≥ K + 2
∫ T L ∏ S ∣ T ∈ B K m R S , T ( − 2 θ S , T ) 2 τ w K d θ ≪ K , d R − 2 L . (14) \int_{\mathbb{T}^L}\prod_{S\mid T\in\mathcal{B}_K}m_R^{S,T}(-2\theta_{S,T})^{2\tau w_K}\,\mathrm{d}\theta\ll_{K,d}R^{-2L}. \tag*{(14)} ∫ T L S ∣ T ∈ B K ∏ m R S , T ( − 2 θ S , T ) 2 τ w K d θ ≪ K , d R − 2 L . (14) Indeed, the covering condition in Finner’s inequality is exactly (10) . Since m R S , T ≤ 1 m_R^{S,T}\le1 m R S , T ≤ 1 2 τ ≥ 2 K + 4 2\tau\ge2K+4 2 τ ≥ 2 K + 4
∫ T L ∏ S ∣ T ∈ B K m R S , T ( − 2 θ S , T ) 2 τ w K d θ ≤ ∏ S ∣ T ∈ B K ( ∫ T ∣ S ∣ ∣ T ∣ m R S , T ( Θ ) 2 τ d Θ ) w K ≪ ∏ S ∣ T ∈ B K R − 2 ∣ S ∣ ∣ T ∣ w K = R − 2 L . \begin{aligned}
\int_{\mathbb{T}^L}\prod_{S\mid T\in\mathcal{B}_K}m_R^{S,T}(-2\theta_{S,T})^{2\tau w_K}\,\mathrm{d}\theta
\le\prod_{S\mid T\in\mathcal{B}_K}\left(\int_{\mathbb{T}^{|S||T|}}m_R^{S,T}(\Theta)^{2\tau}\,\mathrm{d}\Theta\right)^{w_K} \\
&\ll\prod_{S\mid T\in\mathcal{B}_K}R^{-2|S||T|w_K}=R^{-2L}.
\end{aligned} ∫ T L S ∣ T ∈ B K ∏ m R S , T ( − 2 θ S , T ) 2 τ w K d θ ≤ S ∣ T ∈ B K ∏ ( ∫ T ∣ S ∣∣ T ∣ m R S , T ( Θ ) 2 τ d Θ ) w K ≪ S ∣ T ∈ B K ∏ R − 2∣ S ∣∣ T ∣ w K = R − 2 L . For K = 4 K=4 K = 4 2 ∣ 2 2 \mid2 2 ∣ 2
01 ∣ 23 , 02 ∣ 13 , 03 ∣ 12. 01 \mid23,\qquad02 \mid13,\qquad03 \mid12. 01 ∣ 23 , 02 ∣ 13 , 03 ∣ 12.
Each cut sees four edges, every edge crosses exactly two cuts, and w 4 = 1 / 2 w_4=1/2 w 4 = 1/2 Lemma 3.1 is a twelfth-moment estimate, and Finner’s inequality gives the bound
( R − 8 ) 1 / 2 ( R − 8 ) 1 / 2 ( R − 8 ) 1 / 2 = R − 12 = R − 2 ( 4 2 ) (R^{-8})^{1/2}(R^{-8})^{1/2}(R^{-8})^{1/2}=R^{-12}=R^{-2\binom{4}{2}} ( R − 8 ) 1/2 ( R − 8 ) 1/2 ( R − 8 ) 1/2 = R − 12 = R − 2 ( 2 4 ) It is natural to ask whether the lattice-flag count can be replaced by a general circle-method or modular-form estimate. Because (13) is a singular system with low-rank pencils, direct application of the general circle-method estimates in [26 , 13 ] would require taking a trace moment of order proportional to K 2 K^2 K 2 d ≫ K 2 d\gg K^2 d ≫ K 2 [17 ] , but we are unaware of a modular-form estimate that directly supplies the required upper bound for this singular, indefinite matrix equation. The lattice-flag argument exploits its bilinear structure and is what allows the dimension threshold to remain linear in K K K
The major and minor arc estimates
The major arcs.
Fix a suitable parameter 2 ≤ Q ≤ R 2\le Q\le R 2 ≤ Q ≤ R β = a / q + θ \beta=a/q+\theta β = a / q + θ q ≤ 2 Q L q\le2Q^L q ≤ 2 Q L ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 ∥ θ ∥ ∞ ≤ Q / R 2 \|\theta\|_\infty\le Q/R^2 ∥ θ ∥ ∞ ≤ Q / R 2 S a / q + θ S_{a/q+\theta} S a / q + θ
M q , a , θ ( G 0 , … , G K − 1 ) = E r m o d q ∑ u ∈ ( Z d ) K e q ( Q a ( r ) ) e ( Q θ ( u ) ) ∏ i = 0 K − 1 G i ( u i , r i ) , \mathcal{M}_{q,a,\theta}(G_0,\ldots,G_{K-1})=\mathbb{E}_{r\bmod q}\sum_{u\in(\mathbb{Z}^d)^K}e_q(Q_a(r))e(Q_\theta(u))\prod_{i=0}^{K-1}G_i(u_i,r_i), M q , a , θ ( G 0 , … , G K − 1 ) = E r mod q u ∈ ( Z d ) K ∑ e q ( Q a ( r )) e ( Q θ ( u )) i = 0 ∏ K − 1 G i ( u i , r i ) , where each G i G_i G i F i F_i F i μ q \mu_q μ q q q q G i ( u , r ) = ( μ q ∗ F i ) ( y ) G_i(u,r)=(\mu_q*F_i)(y) G i ( u , r ) = ( μ q ∗ F i ) ( y ) y ≡ r ( m o d q ) y\equiv r\pmod q y ≡ r ( mod q ) u − y ∈ { 0 , … , q − 1 } d u-y\in\{0,\ldots,q-1\}^d u − y ∈ { 0 , … , q − 1 } d
Define an operator
K q , a , d S , T : ℓ 2 ( ( Z / q Z ) d T ) ⟶ ℓ 2 ( ( Z / q Z ) d S ) \mathcal{K}^{S,T}_{q,a,d}:\ell^2\left((\mathbb{Z}/q\mathbb{Z})^{dT}\right)\longrightarrow\ell^2\left((\mathbb{Z}/q\mathbb{Z})^{dS}\right) K q , a , d S , T : ℓ 2 ( ( Z / q Z ) d T ) ⟶ ℓ 2 ( ( Z / q Z ) d S ) with normalized counting measure on both spaces, given by
( K q , a , d S , T H ) ( r S ) : = E r T m o d q e q ( − 2 ∑ i ∈ S , j ∈ T a i j r i ⋅ r j ) H ( r T ) . (\mathcal{K}^{S,T}_{q,a,d}H)(r_S):=\mathbb{E}_{r_T\bmod q}e_q\left(-2\sum_{\substack{i\in S,j\in T}}a_{ij}r_i\cdot r_j\right)H(r_T). ( K q , a , d S , T H ) ( r S ) := E r T mod q e q − 2 i ∈ S , j ∈ T ∑ a ij r i ⋅ r j H ( r T ) . By a similar cut reduction, for each cut S ∣ T S\mid T S ∣ T M q , a , θ \mathcal{M}_{q,a,\theta} M q , a , θ K q , a , d S , T ⊗ K P , d S , T ( − 2 θ S , T ) \mathcal{K}^{S,T}_{q,a,d}\otimes\mathcal{K}^{S,T}_{P,d}(-2\theta_{S,T}) K q , a , d S , T ⊗ K P , d S , T ( − 2 θ S , T ) P P P N N N M q , a , θ \mathcal{M}_{q,a,\theta} M q , a , θ
∥ K q , a , d S , T ⊗ K P , d S , T ( − 2 θ S , T ) ∥ 2 → 2 = ∥ K q , a , d S , T ∥ 2 → 2 ∥ K P , d S , T ( − 2 θ S , T ) ∥ 2 → 2 . \left\|\mathcal{K}^{S,T}_{q,a,d}\otimes\mathcal{K}^{S,T}_{P,d}(-2\theta_{S,T})\right\|_{2\to2}
=
\left\|\mathcal{K}^{S,T}_{q,a,d}\right\|_{2\to2}
\left\|\mathcal{K}^{S,T}_{P,d}(-2\theta_{S,T})\right\|_{2\to2}. K q , a , d S , T ⊗ K P , d S , T ( − 2 θ S , T ) 2 → 2 = K q , a , d S , T 2 → 2 K P , d S , T ( − 2 θ S , T ) 2 → 2 . Hence, the major arc analysis reduces to understanding the operator norms of K q , a , d S , T \mathcal{K}^{S,T}_{q,a,d} K q , a , d S , T K P , d S , T ( − 2 θ S , T ) \mathcal{K}^{S,T}_{P,d}(-2\theta_{S,T}) K P , d S , T ( − 2 θ S , T )
∥ K q , a , d S , T ∥ 2 → 2 = ∣ im ( 2 a S ∣ T ) ∣ − d / 2 (15) \left\|\mathcal{K}^{S,T}_{q,a,d}\right\|_{2\to2}
=
\left|\operatorname{im}(2a_{S\mid T})\right|^{-d/2} \tag*{(15)} K q , a , d S , T 2 → 2 = im ( 2 a S ∣ T ) − d /2 (15) where the matrix 2 a S ∣ T = ( 2 a i j ) i ∈ S , j ∈ T 2a_{S|T}=(2a_{ij})_{i\in S,j\in T} 2 a S ∣ T = ( 2 a ij ) i ∈ S , j ∈ T ( Z / q Z ) T → ( Z / q Z ) S (\mathbb{Z}/q\mathbb{Z})^T\to(\mathbb{Z}/q\mathbb{Z})^S ( Z / q Z ) T → ( Z / q Z ) S Corollary 3.2 treats K P , d S , T \mathcal{K}_{P,d}^{S,T} K P , d S , T K q , a , d S , T \mathcal{K}_{q,a,d}^{S,T} K q , a , d S , T
The minor arcs
The minor arcs require an additional pointwise estimate. For ξ ∈ T \xi\in\mathbb{T} ξ ∈ T
K R ( ξ ) = ( e ( ξ r s ) ) r , s ∈ [ R ] , m R ( ξ ) = R − 1 ∥ K R ( ξ ) ∥ 2 → 2 . \mathcal{K}_R(\xi)=\left(e(\xi rs)\right)_{r,s\in[R]},\qquad m_R(\xi)=R^{-1}\lVert\mathcal{K}_R(\xi)\rVert_{2\to2}. K R ( ξ ) = ( e ( ξ r s ) ) r , s ∈ [ R ] , m R ( ξ ) = R − 1 ∥ K R ( ξ ) ∥ 2 → 2 . After isolating the variables attached to a crossing edge e = ( i , j ) e=(i,j) e = ( i , j )
m R S , T ( Θ ) ≤ m R ( Θ i j ) ( i ∈ S , j ∈ T ) . (16) m_R^{S,T}(\Theta)\leq m_R(\Theta_{ij})\qquad(i\in S,\ j\in T). \tag*{(16)} m R S , T ( Θ ) ≤ m R ( Θ ij ) ( i ∈ S , j ∈ T ) . (16) Lemma 3.3. Let 2 ≤ Q ≤ R 2\leq Q\leq R 2 ≤ Q ≤ R ξ ∈ T \xi\in\mathbb{T} ξ ∈ T a / q a/q a / q
1 ≤ q ≤ Q , ∥ ξ − a / q ∥ T ≤ Q q R 2 . 1\leq q\leq Q,\qquad\lVert\xi-a/q\rVert_{\mathbb{T}}\leq\frac{Q}{qR^2}. 1 ≤ q ≤ Q , ∥ ξ − a / q ∥ T ≤ q R 2 Q . Then
m R ( ξ ) 2 ≪ Q − 1 . (17) m_R(\xi)^2\ll Q^{-1}. \tag*{(17)} m R ( ξ ) 2 ≪ Q − 1 . (17) Lemma 3.3 is a Weyl-type estimate; we prove it using the large sieve inequality. Let m Q \mathfrak{m}_Q m Q β ∈ T L \beta\in\mathbb{T}^L β ∈ T L − 2 β e -2\beta_e − 2 β e R R R R ≤ N R\leq N R ≤ N N N N e e e e e e m N S , T ( − 2 β S , T ) ≪ Q − 1 / 2 m_N^{S,T}(-2\beta_{S,T})\ll Q^{-1/2} m N S , T ( − 2 β S , T ) ≪ Q − 1/2
γ : = τ − ( K + 2 ) > 0. \gamma:=\tau-(K+2)>0. γ := τ − ( K + 2 ) > 0. We use the mean value estimate on the first 2 K + 4 2K+4 2 K + 4 2 γ 2\gamma 2 γ (10) give
∫ m Q ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K d β ≪ K , d Q − γ N − 2 L . (18) \int_{\mathfrak{m}_Q}\prod_{S|T\in\mathcal{B}_K}m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K}\,d\beta\ll_{K,d}Q^{-\gamma}N^{-2L}. \tag*{(18)} ∫ m Q S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K d β ≪ K , d Q − γ N − 2 L . (18) Extracting the density increment
It remains to explain why the major arcs retain useful combinatorial information. Let μ q \mu_q μ q q q q
μ q = N q − d 1 q { 0 , … , N q − 1 } d , N q : = ⌊ R q Q L ( L + 2 ) + 3 ⌋ . \mu_q=N_q^{-d}1_{q\{0,\ldots,N_q-1\}^d},\qquad N_q:=\left\lfloor\frac{R}{qQ^{L(L+2)+3}}\right\rfloor. μ q = N q − d 1 q { 0 , … , N q − 1 } d , N q := ⌊ q Q L ( L + 2 ) + 3 R ⌋ . Denote by Λ N m a j \Lambda_N^{\mathrm{maj}} Λ N maj (6) from the major arcs. The major-arc estimate gives, for some η = η ( K , d ) > 1 \eta=\eta(K,d)>1 η = η ( K , d ) > 1
∣ Λ N m a j ( F 0 , … , F K − 1 ) ∣ ≪ Δ , d R ( K − 1 ) d − 2 L ∑ q ≤ 2 Q L q − 1 − η ∏ i = 0 K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) + Q − 2 R K d / 2 − 2 L ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (19) \left|\Lambda_N^{\mathrm{maj}}(F_0,\ldots,F_{K-1})\right|
\ll_{\Delta,d}R^{(K-1)d-2L}\sum_{q\leq2Q^L}q^{-1-\eta}\prod_{i=0}^{K-1}\lVert\mu_q*F_i\rVert_{\ell^K(\mathbb{Z}^d)}
+Q^{-2}R^{Kd/2-2L}\prod_{i=0}^{K-1}\lVert F_i\rVert_{\ell^2(\mathbb{Z}^d)}. \tag*{(19)} Λ N maj ( F 0 , … , F K − 1 ) ≪ Δ , d R ( K − 1 ) d − 2 L q ≤ 2 Q L ∑ q − 1 − η i = 0 ∏ K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) + Q − 2 R K d /2 − 2 L i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (19) The first term is the important one, as μ q ∗ g \mu_q \ast g μ q ∗ g A A A q q q
Returning to (5) , choose Q Q Q α − 1 \alpha^{-1} α − 1 N ≪ Δ , d α − C N \ll_{\Delta,d} \alpha^{-C} N ≪ Δ , d α − C (19) are then negligible. Since
∥ μ q ∗ 1 [ N ] d ∥ ℓ K ( Z d ) ≤ N d / K \left\lVert\mu_q \ast\mathbf{1}_{[N]^d} \right\rVert_{\ell^K(\mathbb{Z}^d)} \le N^{d/K} μ q ∗ 1 [ N ] d ℓ K ( Z d ) ≤ N d / K and ∑ q ≥ 1 q − 1 − η < ∞ \sum_{q \ge1} q^{-1-\eta} < \infty ∑ q ≥ 1 q − 1 − η < ∞ (19) implies that, for some q ≪ Δ , d α − C q \ll_{\Delta,d} \alpha^{-C} q ≪ Δ , d α − C
∑ x ∈ Z d ∣ μ q ∗ g ( x ) ∣ K ≫ Δ , d α K N d . \sum_{x \in\mathbb{Z}^d} \lvert\mu_q \ast g(x) \rvert^K \gg_{\Delta,d} \alpha^K N^d. x ∈ Z d ∑ ∣ μ q ∗ g ( x ) ∣ K ≫ Δ , d α K N d . A standard positive-negative part argument, using μ q ∗ g ≥ − α \mu_q \ast g \ge-\alpha μ q ∗ g ≥ − α ∑ x μ q ∗ g ( x ) = 0 \sum_x \mu_q \ast g(x) = 0 ∑ x μ q ∗ g ( x ) = 0 y ∈ [ 1 + q ( N q − 1 ) , N d ] y \in[1+q(N_q-1),N^d] y ∈ [ 1 + q ( N q − 1 ) , N d ]
( μ q ∗ 1 A ) ( y ) > ( 1 + κ ) α , (\mu_q \ast\mathbf{1}_A)(y) > (1+\kappa)\alpha, ( μ q ∗ 1 A ) ( y ) > ( 1 + κ ) α , where κ > 0 \kappa> 0 κ > 0 Δ \Delta Δ d d d N ≪ Δ , d α − C N \ll_{\Delta,d} \alpha^{-C} N ≪ Δ , d α − C A A A ( 1 + κ ) α (1+\kappa)\alpha ( 1 + κ ) α d d d q ≪ α − C q \ll\alpha^{-C} q ≪ α − C N ′ ≫ α C N N' \gg\alpha^C N N ′ ≫ α C N Δ \Delta Δ O ( 1 + log ( 1 / α ) ) O(1+\log(1/\alpha)) O ( 1 + log ( 1/ α )) O ( 1 + log ( 1 / α ) ) O(1+\log(1/\alpha)) O ( 1 + log ( 1/ α ))
log N ≪ Δ , d ( 1 + log ( 1 / α ) ) 2 , \log N \ll_{\Delta,d} (1+\log(1/\alpha))^2, log N ≪ Δ , d ( 1 + log ( 1/ α ) ) 2 , which is the content of Theorem Theorem 1.1 .
Further directions
The cut operator method may offer a route to quantitative results for more complex configurations. A natural test case is a simplex together with its barycenter, or more general barycentric configurations consisting of the vertices of a simplex together with the barycenters of all of its ℓ \ell ℓ
Organization of the proof
§ 5 introduces the weighted counting form. § 6 develops the cut reduction and proves the mean value estimate. § 7 and § 8 establish the minor- and major-arc estimates. § 9 proves and iterates the density increment. The lattice-flag count underlying Lemma 3.1 is proved in Appendix § 11 .
Notation and standard definitions
We use the following notation throughout the paper.
N = { 1 , 2 , … } \mathbb{N} = \{1,2,\ldots\} N = { 1 , 2 , … } [ N ] = { 1 , … , N } [N] = \{1,\ldots,N\} [ N ] = { 1 , … , N } N N N T = R / Z \mathbb{T} = \mathbb{R}/\mathbb{Z} T = R / Z e ( x ) : = e 2 π i x e(x) := e^{2\pi i x} e ( x ) := e 2 π i x e q ( x ) : = e ( x / q ) e_q(x) := e(x/q) e q ( x ) := e ( x / q )
For x ∈ R x \in\mathbb{R} x ∈ R ∥ x ∥ T : = min n ∈ Z ∣ x − n ∣ \lVert x\rVert_{\mathbb{T}} := \min_{n \in\mathbb{Z}} \lvert x-n\rvert ∥ x ∥ T := min n ∈ Z ∣ x − n ∣ x = ( x 1 , … , x m ) x=(x_1,\ldots,x_m) x = ( x 1 , … , x m ) R m \mathbb{R}^m R m ∥ x ∥ ∞ : = max j ∣ x j ∣ \lVert x\rVert_\infty:= \max_j \lvert x_j\rvert ∥ x ∥ ∞ := max j ∣ x j ∣ x ∈ T m x \in\mathbb{T}^m x ∈ T m ∥ x ∥ ∞ : = max j ∥ x j ∥ T \lVert x\rVert_\infty:= \max_j \lVert x_j\rVert_{\mathbb{T}} ∥ x ∥ ∞ := max j ∥ x j ∥ T diam ∞ ( E ) \operatorname{diam}_\infty(E) diam ∞ ( E )
For functions μ , F : G → C \mu,F:G\to\mathbb{C} μ , F : G → C
( μ ∗ F ) ( y ) : = ∑ x ∈ G μ ( x ) F ( y − x ) . (\mu* F)(y) := \sum_{x\in G} \mu(x)F(y-x). ( μ ∗ F ) ( y ) := x ∈ G ∑ μ ( x ) F ( y − x ) . If T : H 1 → H 2 T:H_1\to H_2 T : H 1 → H 2 T ∗ : H 2 → H 1 T^*:H_2\to H_1 T ∗ : H 2 → H 1 ⟨ T f , g ⟩ H 2 = ⟨ f , T ∗ g ⟩ H 1 \langle Tf,g\rangle_{H_2}=\langle f,T^*g\rangle_{H_1} ⟨ T f , g ⟩ H 2 = ⟨ f , T ∗ g ⟩ H 1
∥ T ∥ 2 → 2 : = sup f ∈ H 1 ∥ f ∥ H 1 = 1 ∥ T f ∥ H 2 . \lVert T\rVert_{2\to2} := \sup_{\substack{f\in H_1\\ \lVert f\rVert_{H_1}=1}} \lVert Tf\rVert_{H_2}. ∥ T ∥ 2 → 2 := f ∈ H 1 ∥ f ∥ H 1 = 1 sup ∥ T f ∥ H 2 . If ( e j ) j (e_j)_j ( e j ) j H 1 H_1 H 1
∥ T ∥ H S : = ( ∑ j ∥ T e j ∥ H 2 2 ) 1 / 2 . \lVert T\rVert_{\mathrm{HS}} := \left(\sum_j \lVert Te_j\rVert_{H_2}^2\right)^{1/2}. ∥ T ∥ HS := ( j ∑ ∥ T e j ∥ H 2 2 ) 1/2 . This value is independent of the choice of orthonormal basis.
If a ∈ ( Z / q Z ) m a\in(\mathbb{Z}/q\mathbb{Z})^m a ∈ ( Z / q Z ) m ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 a a a q q q
We use X ≪ Y X\ll Y X ≪ Y X = O ( Y ) X=O(Y) X = O ( Y ) ∣ X ∣ ≤ C Y \lvert X\rvert\le CY ∣ X ∣ ≤ C Y C > 0 C>0 C > 0 X ≫ Y X\gg Y X ≫ Y Y ≪ X Y\ll X Y ≪ X X ≍ Y X\asymp Y X ≍ Y X ≪ Y X\ll Y X ≪ Y Y ≪ X Y\ll X Y ≪ X
The counting form
Recall the definition of Δ \Delta Δ A ⊆ [ N ] d A\subseteq[N]^d A ⊆ [ N ] d Δ \Delta Δ Δ \Delta Δ
The simplex equations
Put L : = ( K 2 ) L:=\binom{K}{2} L := ( 2 K )
E : = { { i , j } : 0 ≤ i < j ≤ K − 1 } , ∣ E ∣ = L . \mathcal{E}:=\{\{i,j\}:0\le i<j\le K-1\},\qquad\lvert\mathcal{E}\rvert=L. E := {{ i , j } : 0 ≤ i < j ≤ K − 1 } , ∣ E ∣ = L . We count configurations z = ( z 0 , … , z K − 1 ) ∈ ( Z d ) K z=(z_0,\ldots,z_{K-1})\in(\mathbb{Z}^d)^K z = ( z 0 , … , z K − 1 ) ∈ ( Z d ) K
∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( ∀ { i , j } ∈ E ) (20) \lvert z_i-z_j\rvert^2=n\lvert v_i-v_j\rvert^2\qquad(\forall\{i,j\}\in\mathcal{E}) \tag*{(20)} ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( ∀ { i , j } ∈ E ) (20) for some integer n > 0 n>0 n > 0 z 0 , … , z K − 1 z_0,\ldots,z_{K-1} z 0 , … , z K − 1 Δ \Delta Δ n \sqrt{n} n Δ \Delta Δ
For an integer R ≥ 1 R \ge1 R ≥ 1
w R ( n ) : = R − 2 1 { R 2 ≤ n < 2 R 2 } , n ∈ Z . w_R(n) := R^{-2}\mathbf{1}_{\{R^2\le n<2R^2\}}, \qquad n\in\mathbb{Z}. w R ( n ) := R − 2 1 { R 2 ≤ n < 2 R 2 } , n ∈ Z . Thus w R w_R w R n ≍ R 2 n\asymp R^2 n ≍ R 2 ∑ n w R ( n ) = 1 \sum_n w_R(n)=1 ∑ n w R ( n ) = 1
D Δ : = max { i , j } ∈ E ∣ v i − v j ∣ , D 0 : = 1 + ⌊ 2 D Δ ⌋ . D_\Delta:= \max_{\{i,j\}\in\mathcal{E}} |v_i-v_j|,\qquad D_0:=1+\lfloor\sqrt{2}D_\Delta\rfloor. D Δ := { i , j } ∈ E max ∣ v i − v j ∣ , D 0 := 1 + ⌊ 2 D Δ ⌋ . Whenever (20) holds and w R ( n ) ≠ 0 w_R(n)\ne0 w R ( n ) = 0
∣ z i − z j ∣ = n ∣ v i − v j ∣ < 2 R D Δ < D 0 R ( 0 ≤ i < j ≤ K − 1 ) . (21) |z_i-z_j|=\sqrt{n}|v_i-v_j|<\sqrt{2}RD_\Delta<D_0R\qquad(0\le i<j\le K-1). \tag*{(21)} ∣ z i − z j ∣ = n ∣ v i − v j ∣ < 2 R D Δ < D 0 R ( 0 ≤ i < j ≤ K − 1 ) . (21) Fix a constant 0 < σ < 1 / ( 4 D 0 ) 0<\sigma<1/(4D_0) 0 < σ < 1/ ( 4 D 0 ) Δ \Delta Δ N 0 = N 0 ( Δ , d ) ≥ 2 / σ N_0=N_0(\Delta,d)\ge2/\sigma N 0 = N 0 ( Δ , d ) ≥ 2/ σ N ≥ N 0 N\ge N_0 N ≥ N 0
R = ⌊ σ N ⌋ , α = ∣ A ∣ N d . (22) R=\lfloor\sigma N\rfloor,\qquad\alpha=\frac{|A|}{N^d}. \tag*{(22)} R = ⌊ σ N ⌋ , α = N d ∣ A ∣ . (22) The weighted count and the balanced function
For finitely supported functions F 0 , … , F K − 1 : Z d → C F_0,\ldots,F_{K-1}:\mathbb{Z}^d\to\mathbb{C} F 0 , … , F K − 1 : Z d → C
Λ N ( F 0 , … , F K − 1 ) : = ∑ z 0 , … , z K − 1 ∈ Z d ∑ n ∈ Z w R ( n ) 1 { ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( ∀ { i , j } ∈ E ) } ∏ i = 0 K − 1 F i ( z i ) . (23) \Lambda_N(F_0,\ldots,F_{K-1}):=
\sum_{z_0,\ldots,z_{K-1}\in\mathbb{Z}^d}
\sum_{n\in\mathbb{Z}}
w_R(n)\mathbf{1}_{\{|z_i-z_j|^2=n|v_i-v_j|^2\ (\forall\{i,j\}\in\mathcal{E})\}}
\prod_{i=0}^{K-1}F_i(z_i). \tag*{(23)} Λ N ( F 0 , … , F K − 1 ) := z 0 , … , z K − 1 ∈ Z d ∑ n ∈ Z ∑ w R ( n ) 1 { ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( ∀ { i , j } ∈ E )} i = 0 ∏ K − 1 F i ( z i ) . (23) Lemma 5.1 (Kitaoka’s simplex count). If d ≥ 4 K + 4 d\ge4K+4 d ≥ 4 K + 4 N ≥ N 0 N\ge N_0 N ≥ N 0
Λ N ( 1 [ N ] d , … , 1 [ N ] d ) ≫ Δ , d N d R ( K − 1 ) d − 2 L . (24) \Lambda_N(\mathbf{1}_{[N]^d},\ldots,\mathbf{1}_{[N]^d})\gg_{\Delta,d}N^dR^{(K-1)d-2L}. \tag*{(24)} Λ N ( 1 [ N ] d , … , 1 [ N ] d ) ≫ Δ , d N d R ( K − 1 ) d − 2 L . (24) Proof. Put m 0 = 0 m_0=0 m 0 = 0
I N : = { x ∈ [ N ] d : 1 + D 0 R ≤ x j ≤ N − D 0 R for 1 ≤ j ≤ d } . I_N:=\{x\in[N]^d:1+D_0R\le x_j\le N-D_0R\text{ for }1\le j\le d\}. I N := { x ∈ [ N ] d : 1 + D 0 R ≤ x j ≤ N − D 0 R for 1 ≤ j ≤ d } . Since D 0 R < N / 4 D_0R<N/4 D 0 R < N /4 ∣ I N ∣ ≫ d N d |I_N|\gg_d N^d ∣ I N ∣ ≫ d N d
Fix x ∈ I N x\in I_N x ∈ I N z i = x + m i z_i=x+m_i z i = x + m i z i − z j = m i − m j z_i-z_j=m_i-m_j z i − z j = m i − m j
∣ m i − m j ∣ 2 = n ∣ v i − v j ∣ 2 ( { i , j } ∈ E ) . |m_i-m_j|^2=n|v_i-v_j|^2\qquad(\{i,j\}\in\mathcal{E}). ∣ m i − m j ∣ 2 = n ∣ v i − v j ∣ 2 ({ i , j } ∈ E ) . Because m 0 = 0 m_0=0 m 0 = 0
m i ⋅ m j = n ( v i − v 0 ) ⋅ ( v j − v 0 ) ( 1 ≤ i , j ≤ K − 1 ) . m_i\cdot m_j=n(v_i-v_0)\cdot(v_j-v_0)\qquad(1\le i,j\le K-1). m i ⋅ m j = n ( v i − v 0 ) ⋅ ( v j − v 0 ) ( 1 ≤ i , j ≤ K − 1 ) . Kitaoka’s estimate [17 Theorems A–C] , in the form recorded in [22 (2.16) and (3.10)] , therefore gives, whenever R 2 ≤ n < 2 R 2 R^2\le n<2R^2 R 2 ≤ n < 2 R 2
# { ( m 1 , … , m K − 1 ) ∈ ( Z d ) K − 1 : ∣ m i − m j ∣ 2 = n ∣ v i − v j ∣ 2 for every { i , j } ∈ E } ≫ Δ , d n ( K − 1 ) ( d − K ) / 2 . \#\left\{(m_1,\ldots,m_{K-1})\in(\mathbb{Z}^d)^{K-1}:
\begin{array}{c}
|m_i-m_j|^2=n|v_i-v_j|^2\\
\text{for every }\{i,j\}\in\mathcal{E}
\end{array}
\right\}
\gg_{\Delta,d}n^{(K-1)(d-K)/2}. # { ( m 1 , … , m K − 1 ) ∈ ( Z d ) K − 1 : ∣ m i − m j ∣ 2 = n ∣ v i − v j ∣ 2 for every { i , j } ∈ E } ≫ Δ , d n ( K − 1 ) ( d − K ) /2 . By (21) , every such m i m_i m i ∣ m i ∣ < D 0 R |m_i|<D_0R ∣ m i ∣ < D 0 R x + m i ∈ [ N ] d x+m_i\in[N]^d x + m i ∈ [ N ] d x ∈ I N x\in I_N x ∈ I N R 2 R^2 R 2 n n n w R ( n ) = R − 2 w_R(n)=R^{-2} w R ( n ) = R − 2
Λ N ( 1 [ N ] d , … , 1 [ N ] d ) ≫ Δ , d N d ⋅ R 2 ⋅ R − 2 ⋅ R ( K − 1 ) ( d − K ) = N d R ( K − 1 ) d − 2 L = N d R ( K − 1 ) d − 2 L . \Lambda_N(\mathbf{1}_{[N]^d},\ldots,\mathbf{1}_{[N]^d})
\gg_{\Delta,d}N^d\cdot R^2\cdot R^{-2}\cdot R^{(K-1)(d-K)}
=N^dR^{(K-1)d-2L}
=N^dR^{(K-1)d-2L}. Λ N ( 1 [ N ] d , … , 1 [ N ] d ) ≫ Δ , d N d ⋅ R 2 ⋅ R − 2 ⋅ R ( K − 1 ) ( d − K ) = N d R ( K − 1 ) d − 2 L = N d R ( K − 1 ) d − 2 L . Since A A A Δ \Delta Δ
Λ N ( 1 A , … , 1 A ) = 0. (25) \Lambda_N(\mathbf{1}_A,\ldots,\mathbf{1}_A)=0. \tag*{(25)} Λ N ( 1 A , … , 1 A ) = 0. (25) We now extract an important consequence of (25) . There is nothing to prove if A = ∅ A=\varnothing A = ∅ α > 0 \alpha>0 α > 0 g : = 1 A − α 1 [ N ] d g:=\mathbf{1}_A-\alpha\mathbf{1}_{[N]^d} g := 1 A − α 1 [ N ] d
∑ x ∈ Z d g ( x ) = 0 , ∣ g ∣ ≤ 1 , ∥ g ∥ ℓ 2 ( Z d ) 2 = N d α ( 1 − α ) ≤ α N d . (26) \sum_{x\in\mathbb{Z}^d}g(x)=0,\qquad|g|\leq1,\qquad\lVert g\rVert_{\ell^2(\mathbb{Z}^d)}^2=N^d\alpha(1-\alpha)\leq\alpha N^d. \tag*{(26)} x ∈ Z d ∑ g ( x ) = 0 , ∣ g ∣ ≤ 1 , ∥ g ∥ ℓ 2 ( Z d ) 2 = N d α ( 1 − α ) ≤ α N d . (26) For each C ⊂ { 0 , … , K − 1 } C\subset\{0,\ldots,K-1\} C ⊂ { 0 , … , K − 1 } F i C = g F_i^C=g F i C = g i ∈ C i\in C i ∈ C F i C = 1 [ N ] d F_i^C=\mathbf{1}_{[N]^d} F i C = 1 [ N ] d (25) give the exact expansion
0 = ∑ C ⊂ { 0 , … , K − 1 } α K − ∣ C ∣ Λ N ( F 0 C , … , F K − 1 C ) . (27) 0=\sum_{C\subset\{0,\ldots,K-1\}}\alpha^{K-|C|}\Lambda_N(F_0^C,\ldots,F_{K-1}^C). \tag*{(27)} 0 = C ⊂ { 0 , … , K − 1 } ∑ α K − ∣ C ∣ Λ N ( F 0 C , … , F K − 1 C ) . (27) The term with C = ∅ C=\varnothing C = ∅ α K Λ N ( 1 [ N ] d , … , 1 [ N ] d ) \alpha^K\Lambda_N(\mathbf{1}_{[N]^d},\ldots,\mathbf{1}_{[N]^d}) α K Λ N ( 1 [ N ] d , … , 1 [ N ] d ) (24) , we obtain
∑ C ⊂ { 0 , … , K − 1 } C ≠ ∅ α K − ∣ C ∣ ∣ Λ N ( F 0 C , … , F K − 1 C ) ∣ ≫ Δ , d α K N d R ( K − 1 ) d − 2 L . \sum_{\substack{C\subset\{0,\ldots,K-1\}\\ C\neq\varnothing}}\alpha^{K-|C|}\left|\Lambda_N(F_0^C,\ldots,F_{K-1}^C)\right|\gg_{\Delta,d}\alpha^K N^d R^{(K-1)d-2L}. C ⊂ { 0 , … , K − 1 } C = ∅ ∑ α K − ∣ C ∣ Λ N ( F 0 C , … , F K − 1 C ) ≫ Δ , d α K N d R ( K − 1 ) d − 2 L . There are only 2 K − 1 2^K-1 2 K − 1 B ⊂ { 0 , … , K − 1 } \mathcal{B}\subset\{0,\ldots,K-1\} B ⊂ { 0 , … , K − 1 } s : = ∣ B ∣ s:=|\mathcal{B}| s := ∣ B ∣ 1 ≤ s ≤ K 1\leq s\leq K 1 ≤ s ≤ K F i = F i B F_i=F_i^{\mathcal{B}} F i = F i B
∣ Λ N ( F 0 , … , F K − 1 ) ∣ ≫ Δ , d α s N d R ( K − 1 ) d − 2 L , (28) \left|\Lambda_N(F_0,\ldots,F_{K-1})\right|\gg_{\Delta,d}\alpha^sN^dR^{(K-1)d-2L}, \tag*{(28)} ∣ Λ N ( F 0 , … , F K − 1 ) ∣ ≫ Δ , d α s N d R ( K − 1 ) d − 2 L , (28) where F i = g F_i=g F i = g i ∈ B i\in\mathcal{B} i ∈ B F i = 1 [ N ] d F_i=\mathbf{1}_{[N]^d} F i = 1 [ N ] d
Applying the orthogonality relation
For β = ( β i j ) { i , j } ∈ E ∈ T L \beta=(\beta_{ij})_{\{i,j\}\in E}\in\mathbb{T}^L β = ( β ij ) { i , j } ∈ E ∈ T L β j i = β i j \beta_{ji}=\beta_{ij} β j i = β ij u = ( u 0 , … , u K − 1 ) u=(u_0,\ldots,u_{K-1}) u = ( u 0 , … , u K − 1 )
Q β ( u ) : = ∑ { i , j } ∈ E β i j ∣ u i − u j ∣ 2 . Q_\beta(u):=\sum_{\{i,j\}\in E}\beta_{ij}|u_i-u_j|^2. Q β ( u ) := { i , j } ∈ E ∑ β ij ∣ u i − u j ∣ 2 . We apply the orthogonality relation to detect solutions to the L L L Λ N \Lambda_N Λ N ∣ z i − z j ∣ 2 − n ∣ v i − v j ∣ 2 = 0 |z_i-z_j|^2-n|v_i-v_j|^2=0 ∣ z i − z j ∣ 2 − n ∣ v i − v j ∣ 2 = 0 z = ( z 0 , … , z K − 1 ) z=(z_0,\ldots,z_{K-1}) z = ( z 0 , … , z K − 1 ) v = ( v 0 , … , v K − 1 ) v=(v_0,\ldots,v_{K-1}) v = ( v 0 , … , v K − 1 )
∑ { i , j } ∈ E β i j ( ∣ z i − z j ∣ 2 − n ∣ v i − v j ∣ 2 ) = Q β ( z ) − n Q β ( v ) . \sum_{\{i,j\}\in E}\beta_{ij}\left(|z_i-z_j|^2-n|v_i-v_j|^2\right)=Q_\beta(z)-nQ_\beta(v). { i , j } ∈ E ∑ β ij ( ∣ z i − z j ∣ 2 − n ∣ v i − v j ∣ 2 ) = Q β ( z ) − n Q β ( v ) . Thus
1 ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( ∀ { i , j } ∈ E ) = ∫ T L e ( Q β ( z ) − n Q β ( v ) ) d β . \mathbf{1}_{|z_i-z_j|^2=n|v_i-v_j|^2\;(\forall\{i,j\}\in E)}=\int_{\mathbb{T}^L}e\left(Q_\beta(z)-nQ_\beta(v)\right)\,d\beta. 1 ∣ z i − z j ∣ 2 = n ∣ v i − v j ∣ 2 ( ∀ { i , j } ∈ E ) = ∫ T L e ( Q β ( z ) − n Q β ( v ) ) d β . Substituting this identity into the counting form gives
Λ N ( F 0 , … , F K − 1 ) = ∫ T L ∑ n ∈ Z w R ( n ) e ( − n Q β ( v ) ) ∑ z 0 , … , z K − 1 ∈ Z d e ( Q β ( z ) ) ∏ i = 0 K − 1 F i ( z i ) d β . (29) \Lambda_N(F_0,\ldots,F_{K-1})=\int_{\mathbb{T}^L}\sum_{n\in\mathbb{Z}}w_R(n)e\left(-nQ_\beta(v)\right)\sum_{z_0,\ldots,z_{K-1}\in\mathbb{Z}^d}e\left(Q_\beta(z)\right)\prod_{i=0}^{K-1}F_i(z_i)\,d\beta. \tag*{(29)} Λ N ( F 0 , … , F K − 1 ) = ∫ T L n ∈ Z ∑ w R ( n ) e ( − n Q β ( v ) ) z 0 , … , z K − 1 ∈ Z d ∑ e ( Q β ( z ) ) i = 0 ∏ K − 1 F i ( z i ) d β . (29) Observe that the the sum over n n n (29) is w ^ R ( Q β ( v ) ) \widehat{w}_R(Q_\beta(v)) w R ( Q β ( v )) β ∈ T L \beta\in\mathbb{T}^L β ∈ T L
S β ( F 0 , … , F K − 1 ) : = ∑ z 0 , … , z K − 1 ∈ Z d e ( Q β ( z ) ) ∏ i = 0 K − 1 F i ( z i ) . (30) S_\beta(F_0,\ldots,F_{K-1}) := \sum_{z_0,\ldots,z_{K-1}\in\mathbb{Z}^d} e(Q_\beta(z))\prod_{i=0}^{K-1}F_i(z_i). \tag*{(30)} S β ( F 0 , … , F K − 1 ) := z 0 , … , z K − 1 ∈ Z d ∑ e ( Q β ( z )) i = 0 ∏ K − 1 F i ( z i ) . (30) We write S β S_\beta S β F i F_i F i [ N ] d [N]^d [ N ] d (30) is finite and may equivalently be taken over z 0 , … , z K − 1 ∈ [ N ] d z_0,\ldots,z_{K-1}\in[N]^d z 0 , … , z K − 1 ∈ [ N ] d B ⊆ T L \mathcal{B}\subseteq\mathbb{T}^L B ⊆ T L
Λ N B ( F 0 , … , F K − 1 ) : = ∫ B w ^ R ( Q β ( v ) ) S β d β . (31) \Lambda_N^{\mathcal{B}}(F_0,\ldots,F_{K-1}) := \int_{\mathcal{B}}\widehat{w}_R(Q_\beta(v))S_\beta\,d\beta. \tag*{(31)} Λ N B ( F 0 , … , F K − 1 ) := ∫ B w R ( Q β ( v )) S β d β . (31) Taking B = T L \mathcal{B}=\mathbb{T}^L B = T L Λ N = Λ N T L \Lambda_N=\Lambda_N^{\mathbb{T}^L} Λ N = Λ N T L
Since w R w_R w R ∑ n w R ( n ) = 1 \sum_n w_R(n)=1 ∑ n w R ( n ) = 1
∥ w ^ R ∥ L ∞ ( T ) ≤ 1. (32) \|\widehat{w}_R\|_{L^\infty(\mathbb{T})}\leq1. \tag*{(32)} ∥ w R ∥ L ∞ ( T ) ≤ 1. (32) Cut operators
By (31) , both the major arc and minor arc estimates reduce to bounds for the exponential sum
S β ( F 0 , … , F K − 1 ) = ∑ z 0 , … , z K − 1 ∈ [ N ] d e ( Q β ( z ) ) ∏ i = 0 K − 1 F i ( z i ) , S_\beta(F_0,\ldots,F_{K-1})=\sum_{z_0,\ldots,z_{K-1}\in[N]^d}e(Q_\beta(z))\prod_{i=0}^{K-1}F_i(z_i), S β ( F 0 , … , F K − 1 ) = z 0 , … , z K − 1 ∈ [ N ] d ∑ e ( Q β ( z )) i = 0 ∏ K − 1 F i ( z i ) , where the phase is
Q β ( z ) = ∑ { i , j } ∈ E β i j ∣ z i − z j ∣ 2 . Q_\beta(z)=\sum_{\{i,j\}\in E}\beta_{ij}|z_i-z_j|^2. Q β ( z ) = { i , j } ∈ E ∑ β ij ∣ z i − z j ∣ 2 . We estimate this sum by partitioning the variables into two blocks and treating the part of the phase coupling those blocks as an operator kernel.
The cut operator reduction
A cut S ∣ T S\mid T S ∣ T { 0 , … , K − 1 } \{0,\ldots,K-1\} { 0 , … , K − 1 } z = ( z 0 , … , z K − 1 ) z=(z_0,\ldots,z_{K-1}) z = ( z 0 , … , z K − 1 )
z S : = ( z i ) i ∈ S , z T : = ( z j ) j ∈ T . z_S:=(z_i)_{i\in S},\qquad z_T:=(z_j)_{j\in T}. z S := ( z i ) i ∈ S , z T := ( z j ) j ∈ T . An edge contained entirely in S S S T T T Q β ( z ) Q_\beta(z) Q β ( z ) i ∈ S i\in S i ∈ S j ∈ T j\in T j ∈ T
β i j ∣ z i − z j ∣ 2 = β i j ∣ z i ∣ 2 − 2 β i j z i ⋅ z j + β i j ∣ z j ∣ 2 . \beta_{ij}|z_i-z_j|^2=\beta_{ij}|z_i|^2-2\beta_{ij}z_i\cdot z_j+\beta_{ij}|z_j|^2. β ij ∣ z i − z j ∣ 2 = β ij ∣ z i ∣ 2 − 2 β ij z i ⋅ z j + β ij ∣ z j ∣ 2 . Collecting the terms according to the variables on which they depend, define
Q β S ( z S ) : = ∑ { i , i ′ } ∈ E i , i ′ ∈ S β i i ′ ∣ z i − z i ′ ∣ 2 + ∑ i ∈ S , j ∈ T β i j ∣ z i ∣ 2 , Q_\beta^S(z_S):=\sum_{\substack{\{i,i'\}\in E\\ i,i'\in S}}\beta_{ii'}|z_i-z_{i'}|^2+\sum_{\substack{i\in S,j\in T}}\beta_{ij}|z_i|^2, Q β S ( z S ) := { i , i ′ } ∈ E i , i ′ ∈ S ∑ β i i ′ ∣ z i − z i ′ ∣ 2 + i ∈ S , j ∈ T ∑ β ij ∣ z i ∣ 2 , Q β S , T ( z S , z T ) : = − 2 ∑ i ∈ S , j ∈ T β i j z i ⋅ z j , Q_\beta^{S,T}(z_S,z_T):=-2\sum_{\substack{i\in S,j\in T}}\beta_{ij}z_i\cdot z_j, Q β S , T ( z S , z T ) := − 2 i ∈ S , j ∈ T ∑ β ij z i ⋅ z j , Q β T ( z T ) : = ∑ { j , j ′ } ∈ E j , j ′ ∈ T β j j ′ ∣ z j − z j ′ ∣ 2 + ∑ i ∈ S , j ∈ T β i j ∣ z j ∣ 2 . Q_\beta^T(z_T):=\sum_{\substack{\{j,j'\}\in E\\ j,j'\in T}}\beta_{jj'}|z_j-z_{j'}|^2+\sum_{\substack{i\in S,j\in T}}\beta_{ij}|z_j|^2. Q β T ( z T ) := { j , j ′ } ∈ E j , j ′ ∈ T ∑ β j j ′ ∣ z j − z j ′ ∣ 2 + i ∈ S , j ∈ T ∑ β ij ∣ z j ∣ 2 . Then
Q β ( z ) = Q β S ( z S ) + Q β S , T ( z S , z T ) + Q β T ( z T ) . (33) Q_{\beta}(z)=Q_{\beta}^{S}(z_S)+Q_{\beta}^{S,T}(z_S,z_T)+Q_{\beta}^{T}(z_T). \tag*{(33)} Q β ( z ) = Q β S ( z S ) + Q β S , T ( z S , z T ) + Q β T ( z T ) . (33) The first and third terms in (33) depend on variables from only one side of the cut, while the middle term is the only one involving variables from both sides.
We view e ( Q β S , T ( z S , z T ) ) e\left(Q_{\beta}^{S,T}(z_S,z_T)\right) e ( Q β S , T ( z S , z T ) ) z T z_T z T z S z_S z S e ( Q β S ( z S ) ) e\left(Q_{\beta}^{S}(z_S)\right) e ( Q β S ( z S ) ) e ( Q β T ( z T ) ) e\left(Q_{\beta}^{T}(z_T)\right) e ( Q β T ( z T ) ) (33) act by unitary multiplication and therefore do not affect the relevant 2 → 2 2\to2 2 → 2
The cut operator reduction will be applied in two different measure-space settings. On the minor arcs, ( X i , μ i ) = ( [ N ] d , # ) (X_i,\mu_i)=([N]^d,\#) ( X i , μ i ) = ([ N ] d , # ) # \# # X i = B i × ( Z / q Z ) d X_i=B_i\times(\mathbb{Z}/q\mathbb{Z})^d X i = B i × ( Z / q Z ) d B i ⊆ Z d B_i\subseteq\mathbb{Z}^d B i ⊆ Z d X i X_i X i μ i \mu_i μ i
∫ X i f d μ i = E r i m o d q ∑ u i ∈ B i f ( u i , r i ) . \int_{X_i} f\,d\mu_i=\mathbb{E}_{r_i\bmod q}\sum_{u_i\in B_i}f(u_i,r_i). ∫ X i f d μ i = E r i mod q u i ∈ B i ∑ f ( u i , r i ) . Therefore, we formulate the reduction for arbitrary finite measure spaces out of convenience.
For 0 ≤ i ≤ K − 1 0\leq i\leq K-1 0 ≤ i ≤ K − 1 ( X i , μ i ) (X_i,\mu_i) ( X i , μ i )
X S : = ∏ i ∈ S X i , X T : = ∏ j ∈ T X j , μ S : = ⨂ i ∈ S μ i , μ T : = ⨂ j ∈ T μ j . X_S:=\prod_{i\in S}X_i,\qquad X_T:=\prod_{j\in T}X_j,\qquad\mu_S:=\bigotimes_{i\in S}\mu_i,\qquad\mu_T:=\bigotimes_{j\in T}\mu_j. X S := i ∈ S ∏ X i , X T := j ∈ T ∏ X j , μ S := i ∈ S ⨂ μ i , μ T := j ∈ T ⨂ μ j . Given a measurable function Q S , T : X S × X T ⟶ T Q^{S,T}:X_S\times X_T\longrightarrow\mathbb{T} Q S , T : X S × X T ⟶ T
K S , T : L 2 ( X T , μ T ) ⟶ L 2 ( X S , μ S ) , \mathcal{K}^{S,T}:L^2(X_T,\mu_T)\longrightarrow L^2(X_S,\mu_S), K S , T : L 2 ( X T , μ T ) ⟶ L 2 ( X S , μ S ) , ( K S , T h ) ( x S ) : = ∫ X T e ( Q S , T ( x S , x T ) ) h ( x T ) d μ T ( x T ) . (\mathcal{K}^{S,T}h)(x_S):=\int_{X_T}e\left(Q^{S,T}(x_S,x_T)\right)h(x_T)\,d\mu_T(x_T). ( K S , T h ) ( x S ) := ∫ X T e ( Q S , T ( x S , x T ) ) h ( x T ) d μ T ( x T ) . We call every operator of this form a cut operator .
Lemma 6.1 (Cut-operator reduction). Let S ∣ T S\mid T S ∣ T ( X i , μ i ) (X_i,\mu_i) ( X i , μ i ) Q S , T Q^{S,T} Q S , T and K S , T \mathcal{K}^{S,T} K S , T be as above. Let
Q S : X S → T , Q T : X T → T Q^S:X_S\to\mathbb{T},\qquad Q^T:X_T\to\mathbb{T} Q S : X S → T , Q T : X T → T be measurable. Then, for f i ∈ L 2 ( X i , μ i ) f_i\in L^2(X_i,\mu_i) f i ∈ L 2 ( X i , μ i )
∣ ∫ X S ∫ X T e ( Q S ( x S ) + Q S , T ( x S , x T ) + Q T ( x T ) ) ∏ i = 0 K − 1 f i ( x i ) d μ T ( x T ) d μ S ( x S ) ∣ ≤ ∥ K S , T ∥ 2 → 2 ∏ i = 0 K − 1 ∥ f i ∥ L 2 ( X i , μ i ) . (34) \left|\int_{X_S}\int_{X_T}e\left(Q^S(x_S)+Q^{S,T}(x_S,x_T)+Q^T(x_T)\right)\prod_{i=0}^{K-1}f_i(x_i)\,d\mu_T(x_T)\,d\mu_S(x_S)\right|
\leq\left\|\mathcal{K}^{S,T}\right\|_{2\to2}\prod_{i=0}^{K-1}\|f_i\|_{L^2(X_i,\mu_i)}. \tag*{(34)} ∫ X S ∫ X T e ( Q S ( x S ) + Q S , T ( x S , x T ) + Q T ( x T ) ) i = 0 ∏ K − 1 f i ( x i ) d μ T ( x T ) d μ S ( x S ) ≤ K S , T 2 → 2 i = 0 ∏ K − 1 ∥ f i ∥ L 2 ( X i , μ i ) . (34) Proof. Define multiplication operators by
( U S h ) ( x S ) : = e ( Q S ( x S ) ) h ( x S ) , ( U T h ) ( x T ) : = e ( Q T ( x T ) ) h ( x T ) . (U_Sh)(x_S):=e\left(Q^S(x_S)\right)h(x_S),\qquad(U_Th)(x_T):=e\left(Q^T(x_T)\right)h(x_T). ( U S h ) ( x S ) := e ( Q S ( x S ) ) h ( x S ) , ( U T h ) ( x T ) := e ( Q T ( x T ) ) h ( x T ) . Both operators are unitary.
For every h ∈ L 2 ( X T , μ T ) h\in L^2(X_T,\mu_T) h ∈ L 2 ( X T , μ T )
( U S K S , T U T h ) ( x S ) = e ( Q S ( x S ) ) ∫ X T e ( Q S , T ( x S , x T ) ) e ( Q T ( x T ) ) h ( x T ) d μ T ( x T ) = ∫ X T e ( Q S ( x S ) + Q S , T ( x S , x T ) + Q T ( x T ) ) h ( x T ) d μ T ( x T ) . \begin{aligned}
(U_S\mathcal{K}^{S,T}U_Th)(x_S)=e\left(Q^S(x_S)\right)\int_{X_T}e\left(Q^{S,T}(x_S,x_T)\right)e\left(Q^T(x_T)\right)h(x_T)\,d\mu_T(x_T) \\
&=\int_{X_T}e\left(Q^S(x_S)+Q^{S,T}(x_S,x_T)+Q^T(x_T)\right)h(x_T)\,d\mu_T(x_T).
\end{aligned} ( U S K S , T U T h ) ( x S ) = e ( Q S ( x S ) ) ∫ X T e ( Q S , T ( x S , x T ) ) e ( Q T ( x T ) ) h ( x T ) d μ T ( x T ) = ∫ X T e ( Q S ( x S ) + Q S , T ( x S , x T ) + Q T ( x T ) ) h ( x T ) d μ T ( x T ) . Thus U S K S , T U T U_S K^{S,T} U_T U S K S , T U T U S U_S U S U T U_T U T
∥ U S K S , T U T ∥ 2 → 2 = ∥ K S , T ∥ 2 → 2 . \left\|U_S K^{S,T} U_T\right\|_{2\to2}=\left\|K^{S,T}\right\|_{2\to2}. U S K S , T U T 2 → 2 = K S , T 2 → 2 . The integral in the statement can now be written as
∫ X S ( ∏ i ∈ S f i ( x i ) ) ( U S K S , T U T [ ∏ j ∈ T f j ] ) ( x S ) d μ S ( x S ) . \int_{X_S}\left(\prod_{i\in S}f_i(x_i)\right)\left(U_S K^{S,T} U_T\left[\prod_{j\in T}f_j\right]\right)(x_S)\,d\mu_S(x_S). ∫ X S ( i ∈ S ∏ f i ( x i ) ) U S K S , T U T j ∈ T ∏ f j ( x S ) d μ S ( x S ) . Applying Cauchy–Schwarz gives
∥ K S , T ∥ 2 → 2 ∥ ∏ i ∈ S f i ∥ L 2 ( X S , μ S ) ∥ ∏ j ∈ T f j ∥ L 2 ( X T , μ T ) . \left\|K^{S,T}\right\|_{2\to2}\left\|\prod_{i\in S}f_i\right\|_{L^2(X_S,\mu_S)}\left\|\prod_{j\in T}f_j\right\|_{L^2(X_T,\mu_T)}. K S , T 2 → 2 i ∈ S ∏ f i L 2 ( X S , μ S ) j ∈ T ∏ f j L 2 ( X T , μ T ) . The product measures give
∥ ∏ i ∈ S f i ∥ L 2 ( X S , μ S ) = ∏ i ∈ S ∥ f i ∥ L 2 ( X i , μ i ) , ∥ ∏ j ∈ T f j ∥ L 2 ( X T , μ T ) = ∏ j ∈ T ∥ f j ∥ L 2 ( X j , μ j ) , \left\|\prod_{i\in S}f_i\right\|_{L^2(X_S,\mu_S)}=\prod_{i\in S}\|f_i\|_{L^2(X_i,\mu_i)},\qquad\left\|\prod_{j\in T}f_j\right\|_{L^2(X_T,\mu_T)}=\prod_{j\in T}\|f_j\|_{L^2(X_j,\mu_j)}, i ∈ S ∏ f i L 2 ( X S , μ S ) = i ∈ S ∏ ∥ f i ∥ L 2 ( X i , μ i ) , j ∈ T ∏ f j L 2 ( X T , μ T ) = j ∈ T ∏ ∥ f j ∥ L 2 ( X j , μ j ) , which proves the claim.
Tensorization
We now consider a cut operator which naturally appears in the present problem. Take X i = [ R ] d X_i=[R]^d X i = [ R ] d Θ = ( Θ i j ) i ∈ S , j ∈ T ∈ T ∣ S ∣ ∣ T ∣ \Theta=(\Theta_{ij})_{i\in S,j\in T}\in\mathbb{T}^{|S||T|} Θ = ( Θ ij ) i ∈ S , j ∈ T ∈ T ∣ S ∣∣ T ∣ z = ( z 0 , … , z K − 1 ) ∈ ( [ R ] d ) K z=(z_0,\ldots,z_{K-1})\in([R]^d)^K z = ( z 0 , … , z K − 1 ) ∈ ([ R ] d ) K
Q S , T ( z S , z T ) : = ∑ i ∈ S , j ∈ T Θ i j z i ⋅ z j . Q^{S,T}(z_S,z_T):=\sum_{i\in S,j\in T}\Theta_{ij}z_i\cdot z_j. Q S , T ( z S , z T ) := i ∈ S , j ∈ T ∑ Θ ij z i ⋅ z j . The resulting cut operator is
K R , d S , T ( Θ ) : ℓ 2 ( ( [ R ] d ) T ) ⟶ ℓ 2 ( ( [ R ] d ) S ) , K_{R,d}^{S,T}(\Theta):\ell^2(([R]^d)^T)\longrightarrow\ell^2(([R]^d)^S), K R , d S , T ( Θ ) : ℓ 2 (([ R ] d ) T ) ⟶ ℓ 2 (([ R ] d ) S ) , ( K R , d S , T ( Θ ) h ) ( z S ) : = ∑ z T ∈ ( [ R ] d ) T e ( ∑ i ∈ S , j ∈ T Θ i j z i ⋅ z j ) h ( z T ) . (K_{R,d}^{S,T}(\Theta)h)(z_S):=\sum_{z_T\in([R]^d)^T}e\left(\sum_{i\in S,j\in T}\Theta_{ij}z_i\cdot z_j\right)h(z_T). ( K R , d S , T ( Θ ) h ) ( z S ) := z T ∈ ([ R ] d ) T ∑ e i ∈ S , j ∈ T ∑ Θ ij z i ⋅ z j h ( z T ) . We identify this d d d R d ∣ S ∣ × R d ∣ T ∣ R^{d|S|}\times R^{d|T|} R d ∣ S ∣ × R d ∣ T ∣
K R , d S , T ( Θ ) = ( e ( ∑ i ∈ S , j ∈ T Θ i j z i ⋅ z j ) ) z S ∈ ( [ R ] d ) S z T ∈ ( [ R ] d ) T . K_{R,d}^{S,T}(\Theta)=\left(e\left(\sum_{i\in S,j\in T}\Theta_{ij}z_i\cdot z_j\right)\right)_{\substack{z_S\in([R]^d)^S\\z_T\in([R]^d)^T}}. K R , d S , T ( Θ ) = e i ∈ S , j ∈ T ∑ Θ ij z i ⋅ z j z S ∈ ([ R ] d ) S z T ∈ ([ R ] d ) T . For the one-dimensional case, we write K R S , T ( Θ ) : = K R , 1 S , T ( Θ ) K_R^{S,T}(\Theta):=K_{R,1}^{S,T}(\Theta) K R S , T ( Θ ) := K R , 1 S , T ( Θ )
m R S , T ( Θ ) : = R − K / 2 ∥ K R S , T ( Θ ) ∥ 2 → 2 . (35) m_R^{S,T}(\Theta):=R^{-K/2}\left\|K_R^{S,T}(\Theta)\right\|_{2\to2}. \tag*{(35)} m R S , T ( Θ ) := R − K /2 K R S , T ( Θ ) 2 → 2 . (35) By Cauchy–Schwarz, the operator norm of a finite matrix is at most its Hilbert–Schmidt norm. Since the R K R^K R K K R S , T ( Θ ) K_R^{S,T}(\Theta) K R S , T ( Θ )
∥ K R S , T ( Θ ) ∥ 2 → 2 ≤ ∥ K R S , T ( Θ ) ∥ H S = R K / 2 , \left\|K_R^{S,T}(\Theta)\right\|_{2\to2}\leq\left\|K_R^{S,T}(\Theta)\right\|_{\mathrm{HS}}=R^{K/2}, K R S , T ( Θ ) 2 → 2 ≤ K R S , T ( Θ ) HS = R K /2 , and therefore 0 ≤ m R S , T ( Θ ) ≤ 1 0\leq m_R^{S,T}(\Theta)\leq1 0 ≤ m R S , T ( Θ ) ≤ 1
The following lemma gives a relationship between the operator norm of K R S , T K_R^{S,T} K R S , T K R , d S , T K_{R,d}^{S,T} K R , d S , T
Lemma 6.2. For every cut S ∣ T S \mid T S ∣ T
∥ K R , d S , T ( Θ ) ∥ 2 → 2 = ∥ K R S , T ( Θ ) ∥ 2 → 2 d = R K d / 2 m R S , T ( Θ ) d . \left\|\mathcal{K}^{S,T}_{R,d}(\Theta)\right\|_{2\to2}=\left\|\mathcal{K}^{S,T}_{R}(\Theta)\right\|_{2\to2}^{d}=R^{Kd/2}m^{S,T}_{R}(\Theta)^{d}. K R , d S , T ( Θ ) 2 → 2 = K R S , T ( Θ ) 2 → 2 d = R K d /2 m R S , T ( Θ ) d . Proof. Write z i = ( z i , 1 , … , z i , d ) z_i=(z_{i,1},\ldots,z_{i,d}) z i = ( z i , 1 , … , z i , d )
z S , ℓ : = ( z i , ℓ ) i ∈ S ∈ [ R ] S , z T , ℓ : = ( z j , ℓ ) j ∈ T ∈ [ R ] T ( 1 ≤ ℓ ≤ d ) . z_{S,\ell}:=(z_{i,\ell})_{i\in S}\in[R]^S,\qquad z_{T,\ell}:=(z_{j,\ell})_{j\in T}\in[R]^T\qquad(1\leq\ell\leq d). z S , ℓ := ( z i , ℓ ) i ∈ S ∈ [ R ] S , z T , ℓ := ( z j , ℓ ) j ∈ T ∈ [ R ] T ( 1 ≤ ℓ ≤ d ) . Thus the row index z S z_S z S ( z S , 1 , … , z S , d ) (z_{S,1},\ldots,z_{S,d}) ( z S , 1 , … , z S , d ) z T z_T z T ( z T , 1 , … , z T , d ) (z_{T,1},\ldots,z_{T,d}) ( z T , 1 , … , z T , d ) ( z S , z T ) (z_S,z_T) ( z S , z T ) K R , d S , T ( Θ ) \mathcal{K}^{S,T}_{R,d}(\Theta) K R , d S , T ( Θ )
e ( ∑ i ∈ S , j ∈ T Θ i j z i ⋅ z j ) = e ( ∑ ℓ = 1 d ∑ i ∈ S , j ∈ T Θ i j z i , ℓ z j , ℓ ) e\left(\sum_{\substack{i\in S,j\in T}}\Theta_{ij}z_i\cdot z_j\right)=e\left(\sum_{\ell=1}^{d}\sum_{\substack{i\in S,j\in T}}\Theta_{ij}z_{i,\ell}z_{j,\ell}\right) e i ∈ S , j ∈ T ∑ Θ ij z i ⋅ z j = e ℓ = 1 ∑ d i ∈ S , j ∈ T ∑ Θ ij z i , ℓ z j , ℓ = ∏ ℓ = 1 d e ( ∑ i ∈ S , j ∈ T Θ i j z i , ℓ z j , ℓ ) . =\prod_{\ell=1}^{d}e\left(\sum_{\substack{i\in S,j\in T}}\Theta_{ij}z_{i,\ell}z_{j,\ell}\right). = ℓ = 1 ∏ d e i ∈ S , j ∈ T ∑ Θ ij z i , ℓ z j , ℓ . For each ℓ \ell ℓ ( z S , ℓ , z T , ℓ ) (z_{S,\ell},z_{T,\ell}) ( z S , ℓ , z T , ℓ ) K R S , T ( Θ ) \mathcal{K}^{S,T}_{R}(\Theta) K R S , T ( Θ ) d d d
K R , d S , T ( Θ ) = ( K R S , T ( Θ ) ) ⊗ d . \mathcal{K}^{S,T}_{R,d}(\Theta)=\left(\mathcal{K}^{S,T}_{R}(\Theta)\right)^{\otimes d}. K R , d S , T ( Θ ) = ( K R S , T ( Θ ) ) ⊗ d . The standard multiplicativity of the 2 → 2 2\to2 2 → 2
∥ K R , d S , T ( Θ ) ∥ 2 → 2 = ∥ K R S , T ( Θ ) ∥ 2 → 2 d . \left\|\mathcal{K}^{S,T}_{R,d}(\Theta)\right\|_{2\to2}=\left\|\mathcal{K}^{S,T}_{R}(\Theta)\right\|_{2\to2}^{d}. K R , d S , T ( Θ ) 2 → 2 = K R S , T ( Θ ) 2 → 2 d . Finally, by the definition of m R S , T ( Θ ) m^{S,T}_{R}(\Theta) m R S , T ( Θ )
∥ K R S , T ( Θ ) ∥ 2 → 2 d = R K d / 2 m R S , T ( Θ ) d . \left\|\mathcal{K}^{S,T}_{R}(\Theta)\right\|_{2\to2}^{d}=R^{Kd/2}m^{S,T}_{R}(\Theta)^{d}. K R S , T ( Θ ) 2 → 2 d = R K d /2 m R S , T ( Θ ) d . The cut operators K R S , T \mathcal{K}^{S,T}_{R} K R S , T Lemma 6.1 and Lemma 6.2 reduce the analysis to understanding m N S , T ( − 2 β S , T ) m^{S,T}_{N}(-2\beta_{S,T}) m N S , T ( − 2 β S , T ) m N S , T ( − 2 θ S , T ) m^{S,T}_{N}(-2\theta_{S,T}) m N S , T ( − 2 θ S , T ) ∣ im ( 2 a S , T ) ∣ − d / 2 |\operatorname{im}(2a_{S,T})|^{-d/2} ∣ im ( 2 a S , T ) ∣ − d /2 b = ( b e ) e ∈ E b=(b_e)_{e\in\mathcal{E}} b = ( b e ) e ∈ E b i j = b j i : = b { i , j } b_{ij}=b_{ji}:=b_{\{i,j\}} b ij = b j i := b { i , j } b S , T : = ( b i j ) i ∈ S , j ∈ T b_{S,T}:=(b_{ij})_{i\in S,j\in T} b S , T := ( b ij ) i ∈ S , j ∈ T
Reduction to one crossing edge
We next reduce the normalized cut norm to the single-edge norm associated with any one crossing edge. For θ ∈ T \theta\in\mathbb{T} θ ∈ T
K R ( θ ) : = ( e ( θ r s ) ) r , s ∈ [ R ] , m R ( θ ) : = R − 1 ∥ K R ( θ ) ∥ 2 → 2 . \mathcal{K}_{R}(\theta):=\left(e(\theta rs)\right)_{r,s\in[R]},\qquad m_{R}(\theta):=R^{-1}\left\|\mathcal{K}_{R}(\theta)\right\|_{2\to2}. K R ( θ ) := ( e ( θ r s ) ) r , s ∈ [ R ] , m R ( θ ) := R − 1 ∥ K R ( θ ) ∥ 2 → 2 . This is the cut operator associated with a cut having one vertex on each side. Thus K R ( θ ) \mathcal{K}_{R}(\theta) K R ( θ ) R × R R\times R R × R ℓ 2 ( [ R ] ) \ell^2([R]) ℓ 2 ([ R ]) r r r ∑ s ∈ [ R ] e ( θ r s ) h ( s ) \sum_{s\in[R]}e(\theta rs)h(s) ∑ s ∈ [ R ] e ( θ r s ) h ( s ) R R R 0 ≤ m R ( θ ) ≤ 1 0\leq m_R(\theta)\leq1 0 ≤ m R ( θ ) ≤ 1 m R ( − θ ) = m R ( θ ) m_R(-\theta)=m_R(\theta) m R ( − θ ) = m R ( θ )
We shall use the standard block form of Schur’s test: if M = ( M a b ) M=(M_{ab}) M = ( M ab ) p × q p\times q p × q
∥ M ∥ 2 → 2 ≤ p q max a , b ∥ M a b ∥ 2 → 2 . (36) \left\|M\right\|_{2\to2}\leq\sqrt{pq}\max_{a,b}\left\|M_{ab}\right\|_{2\to2}. \tag*{(36)} ∥ M ∥ 2 → 2 ≤ pq a , b max ∥ M ab ∥ 2 → 2 . (36) Lemma 6.3 (Single-edge bound). Let Θ ∈ T ∣ S ∣ ∣ T ∣ \Theta\in\mathbb{T}^{|S||T|} Θ ∈ T ∣ S ∣∣ T ∣ i ∈ S i \in S i ∈ S j ∈ T j \in T j ∈ T m R S , T ( Θ ) ≤ m R ( Θ i j ) m_R^{S,T}(\Theta) \le m_R(\Theta_{ij}) m R S , T ( Θ ) ≤ m R ( Θ ij )
Proof. Abbreviate K : = K R S , T ( Θ ) \mathcal{K} := \mathcal{K}_R^{S,T}(\Theta) K := K R S , T ( Θ )
z ^ S : = ( z k ) k ∈ S ∖ { i } , z ^ T : = ( z ℓ ) ℓ ∈ T ∖ { j } . \widehat{z}_S := (z_k)_{k\in S\setminus\{i\}}, \qquad\widehat{z}_T := (z_\ell)_{\ell\in T\setminus\{j\}}. z S := ( z k ) k ∈ S ∖ { i } , z T := ( z ℓ ) ℓ ∈ T ∖ { j } . For fixed z ^ S \widehat{z}_S z S z ^ T \widehat{z}_T z T
K [ z ^ S , z ^ T ] : = ( e ( Q S , T ( z S , z T ) ) ) z i , z j ∈ [ R ] , \mathcal{K}[\widehat{z}_S,\widehat{z}_T] := \left(e\left(Q^{S,T}(z_S,z_T)\right)\right)_{z_i,z_j\in[R]}, K [ z S , z T ] := ( e ( Q S , T ( z S , z T ) ) ) z i , z j ∈ [ R ] , where the coordinates contained in z ^ S \widehat{z}_S z S z ^ T \widehat{z}_T z T R × R R \times R R × R K \mathcal{K} K z i z_i z i z j z_j z j
Every row is specified by ( z ^ S , z i ) (\widehat{z}_S,z_i) ( z S , z i ) ( z ^ T , z j ) (\widehat{z}_T,z_j) ( z T , z j )
K = ( K [ z ^ S , z ^ T ] ) z ^ S ∈ [ R ] S ∖ { i } , z ^ T ∈ [ R ] T ∖ { j } . \mathcal{K} = \left(\mathcal{K}[\widehat{z}_S,\widehat{z}_T]\right)_{\widehat{z}_S\in[R]^{S\setminus\{i\}},\ \widehat{z}_T\in[R]^{T\setminus\{j\}}}. K = ( K [ z S , z T ] ) z S ∈ [ R ] S ∖ { i } , z T ∈ [ R ] T ∖ { j } . Thus there are R ∣ S ∣ − 1 R^{|S|-1} R ∣ S ∣ − 1 R ∣ T ∣ − 1 R^{|T|-1} R ∣ T ∣ − 1 R × R R \times R R × R Q S , T Q^{S,T} Q S , T
Q S , T ( z S , z T ) = Θ i j z i z j + z i ∑ ℓ ∈ T ∖ { j } Θ i ℓ z ℓ + z j ∑ k ∈ S ∖ { i } Θ k j z k + ∑ k ∈ S ∖ { i } ℓ ∈ T ∖ { j } Θ k ℓ z k z ℓ . (37) Q^{S,T}(z_S,z_T) = \Theta_{ij}z_i z_j + z_i\sum_{\ell\in T\setminus\{j\}}\Theta_{i\ell}z_\ell+ z_j\sum_{k\in S\setminus\{i\}}\Theta_{kj}z_k + \sum_{\substack{k\in S\setminus\{i\}\\ \ell\in T\setminus\{j\}}}\Theta_{k\ell}z_kz_\ell. \tag*{(37)} Q S , T ( z S , z T ) = Θ ij z i z j + z i ℓ ∈ T ∖ { j } ∑ Θ i ℓ z ℓ + z j k ∈ S ∖ { i } ∑ Θ k j z k + k ∈ S ∖ { i } ℓ ∈ T ∖ { j } ∑ Θ k ℓ z k z ℓ . (37) For t ∈ T t \in\mathbb{T} t ∈ T
D R ( t ) : = ( e ( t ) 0 ⋱ 0 e ( R t ) ) . D_R(t) := \begin{pmatrix}
e(t) & 0 & \\
& \ddots& \\
0 & & e(Rt)
\end{pmatrix}. D R ( t ) := e ( t ) 0 0 ⋱ e ( R t ) . Consequently the block has the factorization
K [ z ^ S , z ^ T ] = e ( ∑ k ∈ S ∖ { i } ℓ ∈ T ∖ { j } Θ k ℓ z k z ℓ ) D R ( ∑ ℓ ∈ T ∖ { j } Θ i ℓ z ℓ ) K R ( Θ i j ) D R ( ∑ k ∈ S ∖ { i } Θ k j z k ) . \mathcal{K}[\widehat{z}_S,\widehat{z}_T] = e\left(\sum_{\substack{k\in S\setminus\{i\}\\ \ell\in T\setminus\{j\}}}\Theta_{k\ell}z_kz_\ell\right)D_R\left(\sum_{\ell\in T\setminus\{j\}}\Theta_{i\ell}z_\ell\right)\mathcal{K}_R(\Theta_{ij})D_R\left(\sum_{k\in S\setminus\{i\}}\Theta_{kj}z_k\right). K [ z S , z T ] = e k ∈ S ∖ { i } ℓ ∈ T ∖ { j } ∑ Θ k ℓ z k z ℓ D R ℓ ∈ T ∖ { j } ∑ Θ i ℓ z ℓ K R ( Θ ij ) D R k ∈ S ∖ { i } ∑ Θ k j z k . The scalar factor supplies the last term in (37) . The left and right diagonal factors supply the terms linear in z i z_i z i z j z_j z j K R ( Θ i j ) \mathcal{K}_R(\Theta_{ij}) K R ( Θ ij ) Θ i j z i z j \Theta_{ij}z_i z_j Θ ij z i z j
∥ K [ z ^ S , z ^ T ] ∥ 2 → 2 = ∥ K R ( Θ i j ) ∥ 2 → 2 . \left\|\mathcal{K}[\widehat{z}_S,\widehat{z}_T]\right\|_{2\to2} = \left\|\mathcal{K}_R(\Theta_{ij})\right\|_{2\to2}. ∥ K [ z S , z T ] ∥ 2 → 2 = ∥ K R ( Θ ij ) ∥ 2 → 2 . Applying (36) to the R ∣ S ∣ − 1 × R ∣ T ∣ − 1 R^{|S|-1} \times R^{|T|-1} R ∣ S ∣ − 1 × R ∣ T ∣ − 1
∥ K ∥ 2 → 2 ≤ R ( ∣ S ∣ − 1 ) / 2 R ( ∣ T ∣ − 1 ) / 2 ∥ K R ( Θ i j ) ∥ 2 → 2 = R ( K − 2 ) / 2 ∥ K R ( Θ i j ) ∥ 2 → 2 . \|\mathcal{K}\|_{2\to2} \le R^{(|S|-1)/2}R^{(|T|-1)/2}\left\|\mathcal{K}_R(\Theta_{ij})\right\|_{2\to2} = R^{(K-2)/2}\left\|\mathcal{K}_R(\Theta_{ij})\right\|_{2\to2}. ∥ K ∥ 2 → 2 ≤ R ( ∣ S ∣ − 1 ) /2 R ( ∣ T ∣ − 1 ) /2 ∥ K R ( Θ ij ) ∥ 2 → 2 = R ( K − 2 ) /2 ∥ K R ( Θ ij ) ∥ 2 → 2 . Using the two normalizations now gives
m R S , T ( Θ ) = R − K / 2 ∥ K ∥ 2 → 2 ≤ R − 1 ∥ K R ( Θ i j ) ∥ 2 → 2 = m R ( Θ i j ) . m_R^{S,T}(\Theta) = R^{-K/2}\|\mathcal{K}\|_{2\to2} \le R^{-1}\left\|\mathcal{K}_R(\Theta_{ij})\right\|_{2\to2} = m_R(\Theta_{ij}). m R S , T ( Θ ) = R − K /2 ∥ K ∥ 2 → 2 ≤ R − 1 ∥ K R ( Θ ij ) ∥ 2 → 2 = m R ( Θ ij ) . A high-moment bound
We now prove the one-cut mean-value estimate stated in Lemma 3.1 . Its arithmetic input is the matrix count from Appendix § 11 , which is derived from Kim’s mean-value formula for primitive lattice flags.
Proof of Lemma 3.1 . Abbreviate K : = K R S , T ( Θ ) \mathcal{K} := \mathcal{K}^{S,T}_{R}(\Theta) K := K R S , T ( Θ ) K K ∗ \mathcal{K}\mathcal{K}^{*} K K ∗ λ 1 , … , λ ∣ R ∣ ∣ S ∣ ≥ 0 \lambda_1,\ldots,\lambda_{|R|^{|S|}} \ge0 λ 1 , … , λ ∣ R ∣ ∣ S ∣ ≥ 0
∥ K ∥ 2 → 2 2 K + 4 = ( max a λ a ) K + 2 ≤ ∑ a λ a K + 2 = tr ( ( K K ∗ ) K + 2 ) . (38) \lVert\mathcal{K}\rVert_{2\to2}^{2K+4} = \left(\max_a \lambda_a\right)^{K+2} \le\sum_a \lambda_a^{K+2} = \operatorname{tr}\left((\mathcal{K}\mathcal{K}^{*})^{K+2}\right). \tag*{(38)} ∥ K ∥ 2 → 2 2 K + 4 = ( a max λ a ) K + 2 ≤ a ∑ λ a K + 2 = tr ( ( K K ∗ ) K + 2 ) . (38) In this way we reduce the proof to analyzing tr ( ( K K ∗ ) K + 2 ) \operatorname{tr}\left((\mathcal{K}\mathcal{K}^{*})^{K+2}\right) tr ( ( K K ∗ ) K + 2 )
For z S , z ~ S ∈ [ R ] S z_S,\widetilde{z}_S \in[R]^S z S , z S ∈ [ R ] S ( z S , z ~ S ) (z_S,\widetilde{z}_S) ( z S , z S ) K K ∗ \mathcal{K}\mathcal{K}^{*} K K ∗
( K K ∗ ) z S , z ~ S = ∑ z T ∈ [ R ] T e ( ∑ i ∈ S , j ∈ T Θ i j z i z j ) e ( ∑ i ∈ S , j ∈ T Θ i j z ~ i z j ) ‾ = ∑ z T ∈ [ R ] T e ( ∑ i ∈ S , j ∈ T Θ i j ( z i − z ~ i ) z j ) . (39) \begin{aligned}
(\mathcal{K}\mathcal{K}^{*})_{z_S,\widetilde{z}_S}
=
\sum_{z_T\in[R]^T}
e\left(\sum_{i\in S,j\in T}\Theta_{ij}z_i z_j\right)
\overline{e\left(\sum_{i\in S,j\in T}\Theta_{ij}\widetilde{z}_i z_j\right)} \\
&=
\sum_{z_T\in[R]^T}
e\left(\sum_{i\in S,j\in T}\Theta_{ij}(z_i-\widetilde{z}_i)z_j\right). \tag*{(39)}
\end{aligned} ( K K ∗ ) z S , z S = z T ∈ [ R ] T ∑ e i ∈ S , j ∈ T ∑ Θ ij z i z j e i ∈ S , j ∈ T ∑ Θ ij z i z j = z T ∈ [ R ] T ∑ e i ∈ S , j ∈ T ∑ Θ ij ( z i − z i ) z j . (39) and expanding the trace gives
tr ( ( K K ∗ ) K + 2 ) = ∑ z S ( 1 ) , … , z S ( K + 2 ) ∈ [ R ] S ( K K ∗ ) z S ( 1 ) , z S ( 2 ) ⋯ ( K K ∗ ) z S ( K + 2 ) , z S ( 1 ) . (40) \operatorname{tr}\left((\mathcal{K}\mathcal{K}^{*})^{K+2}\right)
=
\sum_{z_S^{(1)},\ldots,z_S^{(K+2)}\in[R]^S}
(\mathcal{K}\mathcal{K}^{*})_{z_S^{(1)},z_S^{(2)}}\cdots(\mathcal{K}\mathcal{K}^{*})_{z_S^{(K+2)},z_S^{(1)}}. \tag*{(40)} tr ( ( K K ∗ ) K + 2 ) = z S ( 1 ) , … , z S ( K + 2 ) ∈ [ R ] S ∑ ( K K ∗ ) z S ( 1 ) , z S ( 2 ) ⋯ ( K K ∗ ) z S ( K + 2 ) , z S ( 1 ) . (40) Combining (39) with (40) yields
tr ( ( K K ∗ ) K + 2 ) = ∑ z S ( 1 ) , … , z S ( K + 2 ) ∈ [ R ] S z T ( 1 ) , … , z T ( K + 2 ) ∈ [ R ] T e ( ∑ i ∈ S , j ∈ T Θ i j [ ∑ h = 1 K + 1 ( z i ( h ) − z i ( h + 1 ) ) z j ( h ) + ( z i ( K + 2 ) − z i ( 1 ) ) z j ( K + 2 ) ] ) . \operatorname{tr}\left((\mathcal{K}\mathcal{K}^{*})^{K+2}\right)
=
\sum_{\substack{z_S^{(1)},\ldots,z_S^{(K+2)}\in[R]^S\\z_T^{(1)},\ldots,z_T^{(K+2)}\in[R]^T}}
e\left(\sum_{i\in S,j\in T}\Theta_{ij}\left[\sum_{h=1}^{K+1}(z_i^{(h)}-z_i^{(h+1)})z_j^{(h)}+(z_i^{(K+2)}-z_i^{(1)})z_j^{(K+2)}\right]\right). tr ( ( K K ∗ ) K + 2 ) = z S ( 1 ) , … , z S ( K + 2 ) ∈ [ R ] S z T ( 1 ) , … , z T ( K + 2 ) ∈ [ R ] T ∑ e i ∈ S , j ∈ T ∑ Θ ij [ h = 1 ∑ K + 1 ( z i ( h ) − z i ( h + 1 ) ) z j ( h ) + ( z i ( K + 2 ) − z i ( 1 ) ) z j ( K + 2 ) ] . Now integrate each Θ i j \Theta_{ij} Θ ij T \mathbb{T} T
∑ h = 1 K + 1 ( z S ( h ) − z S ( h + 1 ) ) ( z T ( h ) ) T + ( z S ( K + 2 ) − z S ( 1 ) ) ( z T ( K + 2 ) ) T = 0. (41) \sum_{h=1}^{K+1}(z_S^{(h)}-z_S^{(h+1)})(z_T^{(h)})^{\mathsf{T}}+(z_S^{(K+2)}-z_S^{(1)})(z_T^{(K+2)})^{\mathsf{T}}=0. \tag*{(41)} h = 1 ∑ K + 1 ( z S ( h ) − z S ( h + 1 ) ) ( z T ( h ) ) T + ( z S ( K + 2 ) − z S ( 1 ) ) ( z T ( K + 2 ) ) T = 0. (41) Consequently, the integral of the trace is the number of choices z S ( h ) ∈ [ R ] S z_S^{(h)}\in[R]^S z S ( h ) ∈ [ R ] S z T ( h ) ∈ [ R ] T z_T^{(h)}\in[R]^T z T ( h ) ∈ [ R ] T 1 ≤ h ≤ K + 2 1\le h\le K+2 1 ≤ h ≤ K + 2
We next put this constraint into the form required by Lemma B.1 . The differences on the S S S
( z S ( 1 ) − z S ( 2 ) ) + ⋯ + ( z S ( K + 2 ) − z S ( 1 ) ) = 0. (z_S^{(1)}-z_S^{(2)})+\cdots+(z_S^{(K+2)}-z_S^{(1)})=0. ( z S ( 1 ) − z S ( 2 ) ) + ⋯ + ( z S ( K + 2 ) − z S ( 1 ) ) = 0. We may therefore subtract z T ( K + 2 ) z_T^{(K+2)} z T ( K + 2 ) T T T (41) : the change in its left-hand side is the outer product of the zero vector above with ( z T ( K + 2 ) ) T (z_T^{(K+2)})^{\mathsf{T}} ( z T ( K + 2 ) ) T
∑ h = 1 K + 1 ( z S ( h ) − z S ( h + 1 ) ) ( z T ( h ) − z T ( K + 2 ) ) T = 0. \sum_{h=1}^{K+1}(z_S^{(h)}-z_S^{(h+1)})(z_T^{(h)}-z_T^{(K+2)})^{\mathsf{T}}=0. h = 1 ∑ K + 1 ( z S ( h ) − z S ( h + 1 ) ) ( z T ( h ) − z T ( K + 2 ) ) T = 0. For 1 ≤ h ≤ K + 1 1 \le h \le K+1 1 ≤ h ≤ K + 1
U h : = z S ( h ) − z S ( h + 1 ) ∈ Z S , V h : = z T ( h ) − z T ( K + 2 ) ∈ Z T , U_h := z_S^{(h)} - z_S^{(h+1)} \in\mathbb{Z}^S,\qquad V_h := z_T^{(h)} - z_T^{(K+2)} \in\mathbb{Z}^T, U h := z S ( h ) − z S ( h + 1 ) ∈ Z S , V h := z T ( h ) − z T ( K + 2 ) ∈ Z T , and place these vectors in the columns of
U = ( U 1 ⋯ U K + 1 ) ∈ Z ∣ S ∣ × ( K + 1 ) , V = ( V 1 ⋯ V K + 1 ) ∈ Z ∣ T ∣ × ( K + 1 ) . U = (U_1\ \cdots\ U_{K+1}) \in\mathbb{Z}^{|S|\times(K+1)},\qquad V = (V_1\ \cdots\ V_{K+1}) \in\mathbb{Z}^{|T|\times(K+1)}. U = ( U 1 ⋯ U K + 1 ) ∈ Z ∣ S ∣ × ( K + 1 ) , V = ( V 1 ⋯ V K + 1 ) ∈ Z ∣ T ∣ × ( K + 1 ) . Since U V T = ∑ h = 1 K + 1 U h V h T UV^{\mathsf{T}} = \sum_{h=1}^{K+1} U_hV_h^{\mathsf{T}} U V T = ∑ h = 1 K + 1 U h V h T (41) is precisely U V T = 0 UV^{\mathsf{T}} = 0 U V T = 0
Once U U U V V V
z S ( 1 ) , … , z S ( K + 2 ) and z T ( 1 ) , … , z T ( K + 2 ) z_S^{(1)},\ldots,z_S^{(K+2)}\qquad\text{and}\qquad z_T^{(1)},\ldots,z_T^{(K+2)} z S ( 1 ) , … , z S ( K + 2 ) and z T ( 1 ) , … , z T ( K + 2 ) are determined by the two base points z S ( 1 ) ∈ [ R ] S z_S^{(1)} \in[R]^S z S ( 1 ) ∈ [ R ] S z T ( K + 2 ) ∈ [ R ] T z_T^{(K+2)} \in[R]^T z T ( K + 2 ) ∈ [ R ] T
z S ( h + 1 ) = z S ( h ) − U h , z T ( h ) = z T ( K + 2 ) + V h ( 1 ≤ h ≤ K + 1 ) . z_S^{(h+1)} = z_S^{(h)} - U_h,\qquad z_T^{(h)} = z_T^{(K+2)} + V_h\qquad(1 \le h \le K+1). z S ( h + 1 ) = z S ( h ) − U h , z T ( h ) = z T ( K + 2 ) + V h ( 1 ≤ h ≤ K + 1 ) . Thus each pair U , V U,V U , V R ∣ S ∣ R ∣ T ∣ = R K R^{|S|}R^{|T|} = R^K R ∣ S ∣ R ∣ T ∣ = R K [ R ] S [R]^S [ R ] S [ R ] T [R]^T [ R ] T
All entries of U U U V V V [ − R , R ] [-R,R] [ − R , R ] 2 max ( ∣ S ∣ , ∣ T ∣ ) ≤ K + 1 2\max(|S|,|T|) \le K+1 2 max ( ∣ S ∣ , ∣ T ∣ ) ≤ K + 1 ∣ S ∣ + ∣ T ∣ = K < K + 1 |S|+|T|=K<K+1 ∣ S ∣ + ∣ T ∣ = K < K + 1 Lemma B.1 applies with n = K + 1 n=K+1 n = K + 1 U , V U,V U , V
≪ K R ( K + 1 ) ( ∣ S ∣ + ∣ T ∣ ) − 2 ∣ S ∣ ∣ T ∣ = R K ( K + 1 ) − 2 ∣ S ∣ ∣ T ∣ . \ll_K R^{(K+1)(|S|+|T|)-2|S||T|} = R^{K(K+1)-2|S||T|}. ≪ K R ( K + 1 ) ( ∣ S ∣ + ∣ T ∣ ) − 2∣ S ∣∣ T ∣ = R K ( K + 1 ) − 2∣ S ∣∣ T ∣ . Combining this with the R K R^K R K
∫ T ∣ S ∣ ∣ T ∣ tr ( ( K K ∗ ) K + 2 ) d Θ ≪ K R K ( K + 2 ) − 2 ∣ S ∣ ∣ T ∣ . (42) \int_{\mathcal{T}_{|S||T|}} \operatorname{tr}\left((\mathcal{K}\mathcal{K}^{*})^{K+2}\right)\,d\Theta\ll_K R^{K(K+2)-2|S||T|}. \tag*{(42)} ∫ T ∣ S ∣∣ T ∣ tr ( ( K K ∗ ) K + 2 ) d Θ ≪ K R K ( K + 2 ) − 2∣ S ∣∣ T ∣ . (42) Finally, (38) yields
m R S , T ( Θ ) 2 K + 4 = R − K ( K + 2 ) ∥ K ∥ 2 → 2 2 K + 4 ≤ R − K ( K + 2 ) tr ( ( K K ∗ ) K + 2 ) . m_R^{S,T}(\Theta)^{2K+4} = R^{-K(K+2)}\|\mathcal{K}\|_{2\to2}^{2K+4} \le R^{-K(K+2)}\operatorname{tr}\left((\mathcal{K}\mathcal{K}^{*})^{K+2}\right). m R S , T ( Θ ) 2 K + 4 = R − K ( K + 2 ) ∥ K ∥ 2 → 2 2 K + 4 ≤ R − K ( K + 2 ) tr ( ( K K ∗ ) K + 2 ) . Integrating this inequality and applying (42) gives
∫ T ∣ S ∣ ∣ T ∣ m R S , T ( Θ ) 2 K + 4 d Θ ≪ K R − 2 ∣ S ∣ ∣ T ∣ , \int_{\mathcal{T}_{|S||T|}} m_R^{S,T}(\Theta)^{2K+4}\,d\Theta\ll_K R^{-2|S||T|}, ∫ T ∣ S ∣∣ T ∣ m R S , T ( Θ ) 2 K + 4 d Θ ≪ K R − 2∣ S ∣∣ T ∣ , which is (12) .
Balanced cuts and fractional Hölder
Recall that a cut S ∣ T S\mid T S ∣ T balanced if
S\mathbin{\mathaccent\cdot\cup}T=\{0,\ldots,K-1\},\qquad\bigl||S|-|T|\bigr|\le1.
Let B K \mathcal{B}_K B K S ∣ T S\mid T S ∣ T T ∣ S T\mid S T ∣ S B K \mathcal{B}_K B K w K : = L / ( ⌊ K 2 / 4 ⌋ ∣ B K ∣ ) w_K:=L/(\lfloor K^2/4\rfloor|\mathcal{B}_K|) w K := L / (⌊ K 2 /4 ⌋ ∣ B K ∣ )
For a cut S ∣ T S\mid T S ∣ T S ∣ T S\mid T S ∣ T E \mathcal{E} E fractional cover of E \mathcal{E} E
Lemma 6.4. For every edge e ∈ E e \in\mathcal{E} e ∈ E
w K # { S ∣ T ∈ B K : e crosses S ∣ T } = 1. (43) w_K\#\{S \mid T \in\mathcal{B}_K : e\text{ crosses }S \mid T\}=1. \tag*{(43)} w K # { S ∣ T ∈ B K : e crosses S ∣ T } = 1. (43) Proof. Permuting the vertices preserves B K \mathcal{B}_K B K E \mathcal{E} E
Count pairs consisting of a balanced cut and one of its crossing edges. Choosing the cut first gives ∣ B K ∣ ⌊ K 2 / 4 ⌋ |\mathcal{B}_K|\lfloor K^2/4\rfloor ∣ B K ∣ ⌊ K 2 /4 ⌋ ⌊ K 2 / 4 ⌋ \lfloor K^2/4\rfloor ⌊ K 2 /4 ⌋
L # { S ∣ T ∈ B K : e crosses S ∣ T } L\#\{S \mid T \in\mathcal{B}_K : e\text{ crosses }S \mid T\} L # { S ∣ T ∈ B K : e crosses S ∣ T } pairs, since there are L L L
L # { S ∣ T ∈ B K : e crosses S ∣ T } = ∣ B K ∣ ⌊ K 2 / 4 ⌋ . L\#\{S \mid T \in\mathcal{B}_K : e\text{ crosses }S \mid T\}=|\mathcal{B}_K|\lfloor K^2/4\rfloor. L # { S ∣ T ∈ B K : e crosses S ∣ T } = ∣ B K ∣ ⌊ K 2 /4 ⌋ . Multiplying by w K / L w_K/L w K / L (43) .
For K = 4 K=4 K = 4 w 4 = 1 / 2 w_4=1/2 w 4 = 1/2
Figure 1. { 0 , 1 } ∣ { 2 , 3 } \{0,1\}\mid\{2,3\} { 0 , 1 } ∣ { 2 , 3 } { 0 , 2 } ∣ { 1 , 3 } \{0,2\}\mid\{1,3\} { 0 , 2 } ∣ { 1 , 3 } { 0 , 3 } ∣ { 1 , 2 } \{0,3\}\mid\{1,2\} { 0 , 3 } ∣ { 1 , 2 }
The fractional cover identity (6.11) is exactly the hypothesis in the following special case of Finner’s fractional form of Hölder’s inequality [5 ] .
Lemma 6.5 (Fractional Hölder inequality). Let E \mathcal{E} E e ∈ E e \in\mathcal{E} e ∈ E ( X e , μ e ) (X_e,\mu_e) ( X e , μ e ) σ \sigma σ μ : = ⨂ e ∈ E μ e \mu:= \bigotimes_{e\in\mathcal{E}}\mu_e μ := ⨂ e ∈ E μ e J J J 1 ≤ j ≤ J 1 \le j \le J 1 ≤ j ≤ J E j ⊆ E \mathcal{E}_j \subseteq\mathcal{E} E j ⊆ E c j > 0 c_j>0 c j > 0 f j : ∏ e ∈ E j X e ⟶ [ 0 , ∞ ) f_j:\prod_{e\in\mathcal{E}_j}X_e\longrightarrow[0,\infty) f j : ∏ e ∈ E j X e ⟶ [ 0 , ∞ ) f j f_j f j μ j : = ⨂ e ∈ E j μ e \mu_j:=\bigotimes_{e\in\mathcal{E}_j}\mu_e μ j := ⨂ e ∈ E j μ e e ∈ E e\in\mathcal{E} e ∈ E
∑ 1 ≤ j ≤ J e ∈ E j c j = 1. \sum_{\substack{1\le j\le J\\e\in\mathcal{E}_j}}c_j=1. 1 ≤ j ≤ J e ∈ E j ∑ c j = 1. Then, writing x = ( x e ) e ∈ E x=(x_e)_{e\in\mathcal{E}} x = ( x e ) e ∈ E
∫ ∏ j = 1 J f j ( ( x e ) e ∈ E j ) c j d μ ( x ) ≤ ∏ j = 1 J ( ∫ f j d μ j ) c j . \int\prod_{j=1}^{J}f_j\bigl((x_e)_{e\in\mathcal{E}_j}\bigr)^{c_j}\,d\mu(x)\le\prod_{j=1}^{J}\left(\int f_j\,d\mu_j\right)^{c_j}. ∫ j = 1 ∏ J f j ( ( x e ) e ∈ E j ) c j d μ ( x ) ≤ j = 1 ∏ J ( ∫ f j d μ j ) c j . This is [5 Theorem 2.1] , applied with exponents 1 / c j 1/c_j 1/ c j f j c j f_j^{c_j} f j c j
The minor arcs
We begin the section by decomposing T L \mathbb{T}^{L} T L 2 ≤ Q ≤ R 2 \le Q \le R 2 ≤ Q ≤ R
M Q ( 1 ) : = ⋃ q ≤ Q ⋃ a ∈ Z / q Z ( a , q ) = 1 { ξ ∈ T : ∥ ξ − a q ∥ T ≤ Q q R 2 } . (44) \mathfrak{M}_{Q}^{(1)} := \bigcup_{q \le Q}\ \bigcup_{\substack{a \in\mathbb{Z}/q\mathbb{Z}\\(a,q)=1}} \left\{ \xi\in\mathbb{T} : \left\lVert\xi- \frac{a}{q} \right\rVert_{\mathbb{T}} \le\frac{Q}{qR^{2}} \right\}. \tag*{(44)} M Q ( 1 ) := q ≤ Q ⋃ a ∈ Z / q Z ( a , q ) = 1 ⋃ { ξ ∈ T : ξ − q a T ≤ q R 2 Q } . (44) For r ≥ 1 r \ge1 r ≥ 1
M Q ( r ) : = ( M Q ( 1 ) ) r , m Q ( r ) : = T r ∖ M Q ( r ) . \mathfrak{M}_{Q}^{(r)} := \left(\mathfrak{M}_{Q}^{(1)}\right)^{r}, \qquad\mathfrak{m}_{Q}^{(r)} := \mathbb{T}^{r} \setminus\mathfrak{M}_{Q}^{(r)}. M Q ( r ) := ( M Q ( 1 ) ) r , m Q ( r ) := T r ∖ M Q ( r ) . Since the scalar coefficients arising from the cut operators are − 2 β e -2\beta_{e} − 2 β e
M Q : = { β ∈ T L : − 2 β ∈ M Q ( L ) } , m Q : = T L ∖ M Q . \mathfrak{M}_{Q} := \left\{ \beta\in\mathbb{T}^{L} : -2\beta\in\mathfrak{M}_{Q}^{(L)} \right\}, \qquad\mathfrak{m}_{Q} := \mathbb{T}^{L} \setminus\mathfrak{M}_{Q}. M Q := { β ∈ T L : − 2 β ∈ M Q ( L ) } , m Q := T L ∖ M Q . Thus
m Q = ⋃ ε ∈ E { β ∈ T L : − 2 β ε ∈ m Q ( 1 ) } . (45) \mathfrak{m}_{Q} = \bigcup_{\varepsilon\in\mathcal{E}} \left\{ \beta\in\mathbb{T}^{L} : -2\beta_{\varepsilon} \in\mathfrak{m}_{Q}^{(1)} \right\}. \tag*{(45)} m Q = ε ∈ E ⋃ { β ∈ T L : − 2 β ε ∈ m Q ( 1 ) } . (45) The goal of the section is to prove the following minor arc estimate.
Proposition 7.1 (minor arc estimate). Assume 2 ≤ Q ≤ R 2 \le Q \le R 2 ≤ Q ≤ R d ≥ 4 K + 4 d \ge4K + 4 d ≥ 4 K + 4 γ = γ ( K , d ) > 0 \gamma= \gamma(K,d) > 0 γ = γ ( K , d ) > 0 F 0 , … , F K − 1 : Z d → C F_{0},\ldots,F_{K-1} : \mathbb{Z}^{d} \to\mathbb{C} F 0 , … , F K − 1 : Z d → C [ N ] d [N]^{d} [ N ] d
∣ Λ N m Q ( F 0 , … , F K − 1 ) ∣ ≪ σ , Δ , d Q − γ R K d / 2 − 2 L ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (46) \left|\Lambda_{N}^{\mathfrak{m}_{Q}}(F_{0},\ldots,F_{K-1})\right| \ll_{\sigma,\Delta,d} Q^{-\gamma}R^{Kd/2-2L}\prod_{i=0}^{K-1}\lVert F_{i}\rVert_{\ell^{2}(\mathbb{Z}^{d})}. \tag*{(46)} Λ N m Q ( F 0 , … , F K − 1 ) ≪ σ , Δ , d Q − γ R K d /2 − 2 L i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (46) Applying the cut reduction
Since each F i F_i F i [ N ] d [N]^d [ N ] d S β S_{\beta} S β ( [ N ] d ) K ([N]^d)^K ([ N ] d ) K S ∣ T S \mid T S ∣ T X i = [ N ] d X_i = [N]^d X i = [ N ] d f i = F i f_i = F_i f i = F i
Q β S , T ( z S , z T ) = − 2 ∑ i ∈ S , j ∈ T β i j z i ⋅ z j . Q_{\beta}^{S,T}(z_S,z_T) = -2\sum_{i\in S,\,j\in T}\beta_{ij}z_i \cdot z_j. Q β S , T ( z S , z T ) = − 2 i ∈ S , j ∈ T ∑ β ij z i ⋅ z j . so the cut operator in Lemma 6.1 is exactly K N , d S , T ( − 2 β S , T ) \mathcal{K}_{N,d}^{S,T}(-2\beta_{S,T}) K N , d S , T ( − 2 β S , T )
∣ S β ∣ ≤ ∥ K N , d S , T ( − 2 β S , T ) ∥ 2 → 2 ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . |S_{\beta}| \le\left\|\mathcal{K}_{N,d}^{S,T}(-2\beta_{S,T})\right\|_{2\to2}\prod_{i=0}^{K-1}\lVert F_i\rVert_{\ell^{2}(\mathbb{Z}^{d})}. ∣ S β ∣ ≤ K N , d S , T ( − 2 β S , T ) 2 → 2 i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . Applying Lemma 6.2 now gives the required cut-operator estimate directly.
Proposition 7.2 (Cut-operator estimate). For every β ∈ T L \beta\in\mathbb{T}^{L} β ∈ T L S ∣ T S \mid T S ∣ T F 0 , … , F K − 1 : Z d → C F_{0},\ldots,F_{K-1} : \mathbb{Z}^{d} \to\mathbb{C} F 0 , … , F K − 1 : Z d → C [ N ] d [N]^{d} [ N ] d
∣ S β ∣ ≤ N K d / 2 m N S , T ( − 2 β S , T ) d ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (47) |S_{\beta}| \le N^{Kd/2}m_{N}^{S,T}(-2\beta_{S,T})^{d}\prod_{i=0}^{K-1}\lVert F_i\rVert_{\ell^{2}(\mathbb{Z}^{d})}. \tag*{(47)} ∣ S β ∣ ≤ N K d /2 m N S , T ( − 2 β S , T ) d i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (47) :::`
The large-sieve estimate
We now prove the scalar Weyl estimate stated in Lemma 3.3 by applying the analytic large sieve.
Proof of Lemma 3.3. For the proof, write P : = R P := R P := R θ : = ξ \theta:= \xi θ := ξ a , q a,q a , q
1 ≤ q ≤ P 2 Q , ∥ θ − a / q ∥ T ≤ Q q P 2 . 1 \leq q \leq\frac{P^2}{Q}, \qquad\lVert\theta-a/q\rVert_{\mathbb{T}} \leq\frac{Q}{qP^2}. 1 ≤ q ≤ Q P 2 , ∥ θ − a / q ∥ T ≤ q P 2 Q . If q ≤ Q q \leq Q q ≤ Q Q < q ≤ P 2 / Q Q < q \leq P^2/Q Q < q ≤ P 2 / Q q q q ∥ θ − a / q ∥ T ≤ Q / ( q P 2 ) ≤ 1 / q 2 \lVert\theta-a/q\rVert_{\mathbb{T}} \leq Q/(qP^2) \leq1/q^2 ∥ θ − a / q ∥ T ≤ Q / ( q P 2 ) ≤ 1/ q 2
If q < 8 q < 8 q < 8 Q < 8 Q < 8 Q < 8 m P ( θ ) 2 ≤ 1 ≤ 8 Q − 1 m_P(\theta)^2 \leq1 \leq8Q^{-1} m P ( θ ) 2 ≤ 1 ≤ 8 Q − 1 q ≥ 8 q \geq8 q ≥ 8
Partition [ P ] [P] [ P ] j ≪ 1 + P / q j \ll1+P/q j ≪ 1 + P / q I 1 , … , I j I_1,\ldots,I_j I 1 , … , I j q / 8 q/8 q /8 1 ≤ r ≤ j 1 \leq r \leq j 1 ≤ r ≤ j
∣ u − u ′ ∣ ≤ q 8 ( u , u ′ ∈ I r ) . |u-u'| \leq\frac{q}{8} \qquad(u,u' \in I_r). ∣ u − u ′ ∣ ≤ 8 q ( u , u ′ ∈ I r ) . Fix one such interval I r I_r I r u , u ′ ∈ I r u,u' \in I_r u , u ′ ∈ I r ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 ∣ u − u ′ ∣ ≤ q / 8 < q |u-u'| \leq q/8 < q ∣ u − u ′ ∣ ≤ q /8 < q a ( u − u ′ ) a(u-u') a ( u − u ′ ) q q q ∥ a ( u − u ′ ) / q ∥ T ≥ 1 / q \lVert a(u-u')/q\rVert_{\mathbb{T}} \geq1/q ∥ a ( u − u ′ ) / q ∥ T ≥ 1/ q
∣ u − u ′ ∣ ∥ θ − a / q ∥ T ≤ q 8 ⋅ 1 q 2 = 1 8 q . |u-u'|\lVert\theta-a/q\rVert_{\mathbb{T}} \leq\frac{q}{8}\cdot\frac{1}{q^2}=\frac{1}{8q}. ∣ u − u ′ ∣ ∥ θ − a / q ∥ T ≤ 8 q ⋅ q 2 1 = 8 q 1 . It follows from the reverse triangle inequality that
∥ θ u − θ u ′ ∥ T = ∥ θ ( u − u ′ ) ∥ T ≥ ∥ a ( u − u ′ ) q ∥ T − ∥ ( u − u ′ ) ( θ − a q ) ∥ T ≥ ∥ a ( u − u ′ ) q ∥ T − ∣ u − u ′ ∣ ∥ θ − a q ∥ T ≥ 1 q − 1 8 q = 7 8 q . \begin{aligned}
\lVert\theta u-\theta u'\rVert_{\mathbb{T}}
&=\lVert\theta(u-u')\rVert_{\mathbb{T}}\\
&\geq\left\lVert\frac{a(u-u')}{q}\right\rVert_{\mathbb{T}}-\left\lVert(u-u')\left(\theta-\frac{a}{q}\right)\right\rVert_{\mathbb{T}}\\
&\geq\left\lVert\frac{a(u-u')}{q}\right\rVert_{\mathbb{T}}-|u-u'|\left\lVert\theta-\frac{a}{q}\right\rVert_{\mathbb{T}}\\
&\geq\frac{1}{q}-\frac{1}{8q}=\frac{7}{8q}.
\end{aligned} ∥ θ u − θ u ′ ∥ T = ∥ θ ( u − u ′ ) ∥ T ≥ q a ( u − u ′ ) T − ( u − u ′ ) ( θ − q a ) T ≥ q a ( u − u ′ ) T − ∣ u − u ′ ∣ θ − q a T ≥ q 1 − 8 q 1 = 8 q 7 . Thus the points { θ u : u ∈ I r } \{\theta u:u\in I_r\} { θ u : u ∈ I r } T \mathbb{T} T 7 / ( 8 q ) 7/(8q) 7/ ( 8 q )
Applying the analytic large sieve [23 Theorem 1, equation (1.4)] to the trigonometric polynomial ∑ s ∈ [ P ] h ( s ) e ( s x ) \sum_{s\in[P]}h(s)e(sx) ∑ s ∈ [ P ] h ( s ) e ( s x ) { θ u : u ∈ I r } \{\theta u:u\in I_r\} { θ u : u ∈ I r }
∑ u ∈ I r ∣ ∑ s ∈ [ P ] e ( θ u s ) h ( s ) ∣ 2 ≤ ( P + 8 q 7 ) ∥ h ∥ ℓ 2 ( [ P ] ) 2 . \sum_{u\in I_r}\left|\sum_{s\in[P]}e(\theta us)h(s)\right|^2
\leq\left(P+\frac{8q}{7}\right)\lVert h\rVert_{\ell^2([P])}^2. u ∈ I r ∑ s ∈ [ P ] ∑ e ( θ u s ) h ( s ) 2 ≤ ( P + 7 8 q ) ∥ h ∥ ℓ 2 ([ P ]) 2 . Summing this inequality over r ∈ [ j ] r\in[j] r ∈ [ j ] I r I_r I r [ P ] [P] [ P ]
∑ u ∈ [ P ] ∣ ∑ s ∈ [ P ] e ( θ u s ) h ( s ) ∣ 2 ≪ ( 1 + P q ) ( P + q ) ∥ h ∥ ℓ 2 ( [ P ] ) 2 . \sum_{u\in[P]}\left|\sum_{s\in[P]}e(\theta us)h(s)\right|^2
\ll\left(1+\frac{P}{q}\right)(P+q)\lVert h\rVert_{\ell^2([P])}^2. u ∈ [ P ] ∑ s ∈ [ P ] ∑ e ( θ u s ) h ( s ) 2 ≪ ( 1 + q P ) ( P + q ) ∥ h ∥ ℓ 2 ([ P ]) 2 . Taking the supremum over h ∈ ℓ 2 ( [ P ] ) h\in\ell^2([P]) h ∈ ℓ 2 ([ P ]) ∥ h ∥ ℓ 2 ( [ P ] ) ≤ 1 \lVert h\rVert_{\ell^2([P])}\leq1 ∥ h ∥ ℓ 2 ([ P ]) ≤ 1
∥ K P ( θ ) ∥ 2 → 2 2 ≪ ( 1 + P q ) ( P + q ) . \lVert K_P(\theta)\rVert_{2\to2}^2 \ll\left(1+\frac{P}{q}\right)(P+q). ∥ K P ( θ ) ∥ 2 → 2 2 ≪ ( 1 + q P ) ( P + q ) . Since m P ( θ ) = P − 1 ∥ K P ( θ ) ∥ 2 → 2 m_P(\theta)=P^{-1}\lVert K_P(\theta)\rVert_{2\to2} m P ( θ ) = P − 1 ∥ K P ( θ ) ∥ 2 → 2
m P ( θ ) 2 ≪ P − 2 ( 1 + P q ) ( P + q ) ≪ q − 1 + P − 1 + q P − 2 ≪ Q − 1 . m_P(\theta)^2 \ll P^{-2}\left(1+\frac{P}{q}\right)(P+q)\ll q^{-1}+P^{-1}+qP^{-2}\ll Q^{-1}. m P ( θ ) 2 ≪ P − 2 ( 1 + q P ) ( P + q ) ≪ q − 1 + P − 1 + q P − 2 ≪ Q − 1 . Here we used
q > Q , Q ≤ P , q ≤ P 2 Q . q>Q,\qquad Q\le P,\qquad q\le\frac{P^2}{Q}. q > Q , Q ≤ P , q ≤ Q P 2 . Proof of the minor arc estimate
We now prove the minor arc estimate by combining fractional Hölder, the large-sieve estimate, and the high moment estimate.
Proof of Proposition 7.1. Put τ : = d ⌊ K 2 / 4 ⌋ / ( 2 L ) \tau:=d\lfloor K^2/4\rfloor/(2L) τ := d ⌊ K 2 /4 ⌋ / ( 2 L ) 2 τ w K = d / ∣ B K ∣ 2\tau w_K=d/\lvert\mathcal{B}_K\rvert 2 τ w K = d / ∣ B K ∣
∣ S β ∣ ≤ N K d / 2 ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (48) \lvert S_\beta\rvert\le N^{Kd/2}\prod_{S\mid T\in\mathcal{B}_K}m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K}\prod_{i=0}^{K-1}\lVert F_i\rVert_{\ell^2(\mathbb{Z}_d)}. \tag*{(48)} ∣ S β ∣ ≤ N K d /2 S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (48) The dimension hypothesis gives
τ ≥ ( 4 K + 4 ) ( K 2 − 1 ) 8 L = ( K + 1 ) 2 K > K + 2. \tau\ge\frac{(4K+4)(K^2-1)}{8L}=\frac{(K+1)^2}{K}>K+2. τ ≥ 8 L ( 4 K + 4 ) ( K 2 − 1 ) = K ( K + 1 ) 2 > K + 2. Set γ : = τ − ( K + 2 ) > 0 \gamma:=\tau-(K+2)>0 γ := τ − ( K + 2 ) > 0 (31) , (32) , and (48) , it remains to prove
∫ m Q ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K d β ≪ K , d Q − γ N − 2 L . (49) \int_{\mathfrak{m}_Q}\prod_{S\mid T\in\mathcal{B}_K}m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K}\,d\beta\ll_{K,d}Q^{-\gamma}N^{-2L}. \tag*{(49)} ∫ m Q S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K d β ≪ K , d Q − γ N − 2 L . (49) For each e ∈ E e\in\mathcal{E} e ∈ E
I e : = ∫ T L 1 m Q ( 1 ) ( − 2 β e ) ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K d β . I_e:=\int_{\mathbb{T}_L}\mathbf{1}_{\mathfrak{m}_Q}^{(1)}(-2\beta_e)\prod_{S\mid T\in\mathcal{B}_K}m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K}\,d\beta. I e := ∫ T L 1 m Q ( 1 ) ( − 2 β e ) S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K d β . By (45) and nonnegativity,
∫ m Q ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K d β ≤ ∑ e ∈ E I e . \int_{\mathfrak{m}_Q}\prod_{S\mid T\in\mathcal{B}_K}m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K}\,d\beta\le\sum_{e\in\mathcal{E}}I_e. ∫ m Q S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K d β ≤ e ∈ E ∑ I e . It therefore suffices to prove
I e ≪ K , d Q − γ N − 2 L (50) I_e\ll_{K,d}Q^{-\gamma}N^{-2L} \tag*{(50)} I e ≪ K , d Q − γ N − 2 L (50) for every e ∈ E e\in\mathcal{E} e ∈ E
Fix e ∈ E e\in\mathcal{E} e ∈ E J = ∣ B K ∣ J=\lvert\mathcal{B}_K\rvert J = ∣ B K ∣ 1 , … , J 1,\ldots,J 1 , … , J S ∣ T ∈ B K S\mid T\in\mathcal{B}_K S ∣ T ∈ B K j j j S ∣ T S\mid T S ∣ T E j \mathcal{E}_j E j S ∣ T S\mid T S ∣ T c j c_j c j w K w_K w K f S , T f_{S,T} f S , T f j f_j f j
For each balanced cut S ∣ T S\mid T S ∣ T
f S , T ( β S , T ) : = { 1 m Q ( 1 ) ( − 2 β e ) m N S , T ( − 2 β S , T ) 2 τ , e crosses S ∣ T , m N S , T ( − 2 β S , T ) 2 τ , e does not cross S ∣ T . f_{S,T}(\beta_{S,T}):=
\begin{cases}
\mathbf{1}_{\mathfrak{m}_Q}^{(1)}(-2\beta_e)m_N^{S,T}(-2\beta_{S,T})^{2\tau}, & e\text{ crosses }S\mid T,\\
m_N^{S,T}(-2\beta_{S,T})^{2\tau}, & e\text{ does not cross }S\mid T.
\end{cases} f S , T ( β S , T ) := { 1 m Q ( 1 ) ( − 2 β e ) m N S , T ( − 2 β S , T ) 2 τ , m N S , T ( − 2 β S , T ) 2 τ , e crosses S ∣ T , e does not cross S ∣ T . When e e e S ∣ T S \mid T S ∣ T β e \beta_e β e β S , T \beta_{S,T} β S , T E j \mathcal{E}_j E j
For every e ′ ∈ E e' \in\mathcal{E} e ′ ∈ E
∑ S ∣ T ∈ B K e ′ crosses S ∣ T w K = 1 , \sum_{\substack{S\mid T\in\mathcal{B}_K\\ e'\text{ crosses }S\mid T}} w_K = 1, S ∣ T ∈ B K e ′ crosses S ∣ T ∑ w K = 1 , which is exactly the hypothesis of Lemma 6.5. Moreover,
∏ S ∣ T ∈ B K f S , T ( β S , T ) w K = ( ∏ S ∣ T ∈ B K e crosses S ∣ T 1 m Q ( 1 ) ( − 2 β e ) w K ) ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K = 1 m Q ( 1 ) ( − 2 β e ) ∏ S ∣ T ∈ B K m N S , T ( − 2 β S , T ) 2 τ w K , \begin{aligned}
\prod_{S\mid T\in\mathcal{B}_K} f_{S,T}(\beta_{S,T})^{w_K}
=
\left(\prod_{\substack{S\mid T\in\mathcal{B}_K\\ e\text{ crosses }S\mid T}} \mathbf{1}_{m_Q^{(1)}}(-2\beta_e)^{w_K}\right)
\prod_{S\mid T\in\mathcal{B}_K} m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K} \\
&= \mathbf{1}_{m_Q^{(1)}}(-2\beta_e)
\prod_{S\mid T\in\mathcal{B}_K} m_N^{S,T}(-2\beta_{S,T})^{2\tau w_K},
\end{aligned} S ∣ T ∈ B K ∏ f S , T ( β S , T ) w K = S ∣ T ∈ B K e crosses S ∣ T ∏ 1 m Q ( 1 ) ( − 2 β e ) w K S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K = 1 m Q ( 1 ) ( − 2 β e ) S ∣ T ∈ B K ∏ m N S , T ( − 2 β S , T ) 2 τ w K , where the last equality follows from (43) . Hence
I e = ∫ T L ∏ S ∣ T ∈ B K f S , T ( β S , T ) w K d β . I_e=\int_{\mathbb{T}^{L}}\prod_{S\mid T\in\mathcal{B}_K}f_{S,T}(\beta_{S,T})^{w_K}\,d\beta. I e = ∫ T L S ∣ T ∈ B K ∏ f S , T ( β S , T ) w K d β . Lemma 6.5 therefore gives
I e ≤ ∏ S ∣ T ∈ B K e crosses S ∣ T ( ∫ T ∣ S ∣ ∣ T ∣ 1 m Q ( 1 ) ( − 2 β e ) m N S , T ( − 2 β S , T ) 2 τ d β S , T ) w K × ∏ S ∣ T ∈ B K e does not cross S ∣ T ( ∫ T ∣ S ∣ ∣ T ∣ m N S , T ( − 2 β S , T ) 2 τ d β S , T ) w K . (51) \begin{aligned}
I_e\leq
\prod_{\substack{S\mid T\in\mathcal{B}_K\\ e\text{ crosses }S\mid T}}
\left(\int_{\mathbb{T}^{|S||T|}}\mathbf{1}_{m_Q^{(1)}}(-2\beta_e)m_N^{S,T}(-2\beta_{S,T})^{2\tau}\,d\beta_{S,T}\right)^{w_K} \\
&\times
\prod_{\substack{S\mid T\in\mathcal{B}_K\\ e\text{ does not cross }S\mid T}}
\left(\int_{\mathbb{T}^{|S||T|}}m_N^{S,T}(-2\beta_{S,T})^{2\tau}\,d\beta_{S,T}\right)^{w_K}. \tag*{(51)}
\end{aligned} I e ≤ S ∣ T ∈ B K e crosses S ∣ T ∏ ( ∫ T ∣ S ∣∣ T ∣ 1 m Q ( 1 ) ( − 2 β e ) m N S , T ( − 2 β S , T ) 2 τ d β S , T ) w K × S ∣ T ∈ B K e does not cross S ∣ T ∏ ( ∫ T ∣ S ∣∣ T ∣ m N S , T ( − 2 β S , T ) 2 τ d β S , T ) w K . (51) We now estimate the integrals on the right. Fix a balanced cut S ∣ T S \mid T S ∣ T
( β i j ) i ∈ S , j ∈ T ⟼ ( − 2 β i j ) i ∈ S , j ∈ T (\beta_{ij})_{i\in S,j\in T}\longmapsto(-2\beta_{ij})_{i\in S,j\in T} ( β ij ) i ∈ S , j ∈ T ⟼ ( − 2 β ij ) i ∈ S , j ∈ T preserves normalized Haar measure on T ∣ S ∣ ∣ T ∣ \mathbb{T}^{|S||T|} T ∣ S ∣∣ T ∣ Lemma 3.1 gives
∫ T ∣ S ∣ ∣ T ∣ m N S , T ( − 2 β S , T ) 2 K + 4 d β S , T ≪ K N − 2 ∣ S ∣ ∣ T ∣ . (52) \int_{\mathbb{T}^{|S||T|}}m_N^{S,T}(-2\beta_{S,T})^{2K+4}\,d\beta_{S,T}\ll_K N^{-2|S||T|}. \tag*{(52)} ∫ T ∣ S ∣∣ T ∣ m N S , T ( − 2 β S , T ) 2 K + 4 d β S , T ≪ K N − 2∣ S ∣∣ T ∣ . (52) Since 0 ≤ m N S , T ( − 2 β S , T ) ≤ 1 0\leq m_N^{S,T}(-2\beta_{S,T})\leq1 0 ≤ m N S , T ( − 2 β S , T ) ≤ 1 2 τ > 2 K + 4 2\tau>2K+4 2 τ > 2 K + 4 (52) gives
∫ T ∣ S ∣ ∣ T ∣ m N S , T ( − 2 β S , T ) 2 τ d β S , T ≪ K N − 2 ∣ S ∣ ∣ T ∣ . (53) \int_{\mathbb{T}^{|S||T|}}m_N^{S,T}(-2\beta_{S,T})^{2\tau}\,d\beta_{S,T}\ll_K N^{-2|S||T|}. \tag*{(53)} ∫ T ∣ S ∣∣ T ∣ m N S , T ( − 2 β S , T ) 2 τ d β S , T ≪ K N − 2∣ S ∣∣ T ∣ . (53) Suppose that e e e S ∣ T S\mid T S ∣ T − 2 β e ∈ m Q ( 1 ) -2\beta_e\in m_Q^{(1)} − 2 β e ∈ m Q ( 1 ) m Q ( 1 ) m_Q^{(1)} m Q ( 1 ) N ≥ R N\geq R N ≥ R − 2 β e -2\beta_e − 2 β e P = N P=N P = N Q / ( q N 2 ) ≤ Q / ( q R 2 ) Q/(qN^2)\leq Q/(qR^2) Q / ( q N 2 ) ≤ Q / ( q R 2 ) Lemma 6.3 gives
m N S , T ( − 2 β S , T ) ≤ m N ( − 2 β e ) ≪ Q − 1 / 2 . m_N^{S,T}(-2\beta_{S,T})\leq m_N(-2\beta_e)\ll Q^{-1/2}. m N S , T ( − 2 β S , T ) ≤ m N ( − 2 β e ) ≪ Q − 1/2 . Since 2 τ = 2 K + 4 + 2 γ 2\tau= 2K + 4 + 2\gamma 2 τ = 2 K + 4 + 2 γ
m N S , T ( − 2 β S , T ) 2 τ ≪ K , d Q − γ m N S , T ( − 2 β S , T ) 2 K + 4 . m_N^{S,T}(-2\beta_{S,T})^{2\tau} \ll_{K,d} Q^{-\gamma}m_N^{S,T}(-2\beta_{S,T})^{2K+4}. m N S , T ( − 2 β S , T ) 2 τ ≪ K , d Q − γ m N S , T ( − 2 β S , T ) 2 K + 4 . Using (52) , we obtain
∫ T ∣ S ∣ ∣ T ∣ 1 q ( 1 ) ( − 2 β e ) m N S , T ( − 2 β S , T ) 2 τ d β S , T ≪ K , d Q − γ N − 2 ∣ S ∣ ∣ T ∣ . (54) \int_{\mathbb{T}^{\lvert S\rvert\lvert T\rvert}}1_{\mathfrak{q}}^{(1)}(-2\beta_e)m_N^{S,T}(-2\beta_{S,T})^{2\tau}\,d\beta_{S,T}\ll_{K,d} Q^{-\gamma}N^{-2\lvert S\rvert\lvert T\rvert}. \tag*{(54)} ∫ T ∣ S ∣ ∣ T ∣ 1 q ( 1 ) ( − 2 β e ) m N S , T ( − 2 β S , T ) 2 τ d β S , T ≪ K , d Q − γ N − 2 ∣ S ∣ ∣ T ∣ . (54) For the cuts crossed by e e e (54) contributes both a factor Q − γ w K Q^{-\gamma w_K} Q − γ w K N − 2 w K ∣ S ∣ ∣ T ∣ N^{-2w_K\lvert S\rvert\lvert T\rvert} N − 2 w K ∣ S ∣ ∣ T ∣ (51) . For the remaining cuts, (53) contributes the factor N − 2 w K ∣ S ∣ ∣ T ∣ N^{-2w_K\lvert S\rvert\lvert T\rvert} N − 2 w K ∣ S ∣ ∣ T ∣
I e ≪ K , d Q − γ w K # { S ∣ T ∈ B K : e crosses S ∣ T } N − 2 w K ∑ S ∣ T ∈ B K ∣ S ∣ ∣ T ∣ . I_e\ll_{K,d}Q^{-\gamma w_K\#\{S\mid T\in\mathcal{B}_K:e\text{ crosses }S\mid T\}}N^{-2w_K\sum_{S\mid T\in\mathcal{B}_K}\lvert S\rvert\lvert T\rvert}. I e ≪ K , d Q − γ w K # { S ∣ T ∈ B K : e crosses S ∣ T } N − 2 w K ∑ S ∣ T ∈ B K ∣ S ∣ ∣ T ∣ . The fractional-cover identity (43) gives
w K # { S ∣ T ∈ B K : e crosses S ∣ T } = 1. w_K\#\{S\mid T\in\mathcal{B}_K:e\text{ crosses }S\mid T\}=1. w K # { S ∣ T ∈ B K : e crosses S ∣ T } = 1. Moreover, every balanced cut has ∣ S ∣ ∣ T ∣ = ⌊ K 2 / 4 ⌋ \lvert S\rvert\lvert T\rvert=\lfloor K^2/4\rfloor ∣ S ∣ ∣ T ∣ = ⌊ K 2 /4 ⌋ w K w_K w K
w K ∑ S ∣ T ∈ B K ∣ S ∣ ∣ T ∣ = w K ∣ B K ∣ ⌊ K 2 / 4 ⌋ = L . (55) w_K\sum_{S\mid T\in\mathcal{B}_K}\lvert S\rvert\lvert T\rvert=w_K\lvert\mathcal{B}_K\rvert\lfloor K^2/4\rfloor=L. \tag*{(55)} w K S ∣ T ∈ B K ∑ ∣ S ∣ ∣ T ∣ = w K ∣ B K ∣ ⌊ K 2 /4 ⌋ = L . (55) Putting these facts together proves (50) , and hence (49) . Combining that estimate with (31) , (32) , and (48) gives
∣ Λ N m Q ( F 0 , … , F K − 1 ) ∣ ≪ Δ , d Q − γ N K d / 2 − 2 L ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . \lvert\Lambda_N^{m_Q}(F_0,\ldots,F_{K-1})\rvert\ll_{\Delta,d}Q^{-\gamma}N^{Kd/2-2L}\prod_{i=0}^{K-1}\lVert F_i\rVert_{\ell^2(\mathbb{Z}^d)}. ∣ Λ N m Q ( F 0 , … , F K − 1 )∣ ≪ Δ , d Q − γ N K d /2 − 2 L i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . Finally, N ≥ 2 / σ N\ge2/\sigma N ≥ 2/ σ R = ⌊ σ N ⌋ ≥ σ N / 2 R=\lfloor\sigma N\rfloor\ge\sigma N/2 R = ⌊ σ N ⌋ ≥ σ N /2 (46) follows as well. □
The major arcs
Throughout this section assume
R ≥ 4 Q L ( L + 3 ) + 3 . (56) R\ge4Q^{L(L+3)+3}. \tag*{(56)} R ≥ 4 Q L ( L + 3 ) + 3 . (56) For 1 ≤ q ≤ 2 Q L 1\le q\le2Q^L 1 ≤ q ≤ 2 Q L
N q : = ⌊ R q Q L ( L + 2 ) + 3 ⌋ . (57) N_q:=\left\lfloor\frac{R}{qQ^{L(L+2)+3}}\right\rfloor. \tag*{(57)} N q := ⌊ q Q L ( L + 2 ) + 3 R ⌋ . (57) Since q ≤ 2 Q L q\le2Q^L q ≤ 2 Q L (56) gives N q ≥ 1 N_q\ge1 N q ≥ 1 μ q \mu_q μ q Z d \mathbb{Z}^d Z d
μ q ( x ) : = 1 N q d 1 { 0 , … , N q − 1 } d ( x ) . \mu_q(x):=\frac{1}{N_q^d}1_{\{0,\ldots,N_q-1\}^d}(x). μ q ( x ) := N q d 1 1 { 0 , … , N q − 1 } d ( x ) . Thus, for every finitely supported F : Z d → C F:\mathbb{Z}^d\to\mathbb{C} F : Z d → C
( μ q ∗ F ) ( y ) = 1 N q d ∑ h ∈ { 0 , … , N q − 1 } d F ( y − q h ) . (\mu_q*F)(y)=\frac{1}{N_q^d}\sum_{h\in\{0,\ldots,N_q-1\}^d}F(y-qh). ( μ q ∗ F ) ( y ) = N q d 1 h ∈ { 0 , … , N q − 1 } d ∑ F ( y − q h ) . The goal of the section is to prove the following major arc estimate.
Proposition 8.1 (major arc estimate). Assume 2 ≤ Q ≤ R 2 \le Q \le R 2 ≤ Q ≤ R d ≥ 4 K + 4 d \ge4K+4 d ≥ 4 K + 4 (56) . There is η = η ( K , d ) > 1 \eta=\eta(K,d)>1 η = η ( K , d ) > 1 F 0 , … , F K − 1 : Z d → C F_0,\ldots,F_{K-1}:\mathbb{Z}^d\to\mathbb{C} F 0 , … , F K − 1 : Z d → C [ N ] d [N]^d [ N ] d
∣ Λ N M Q ( F 0 , … , F K − 1 ) ∣ ≪ Δ , d , σ R ( K − 1 ) d − 2 L ∑ q ≤ 2 Q L q − 1 − η ∏ i = 0 K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) + Q − 2 R K d / 2 − 2 L ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (58) \left|\Lambda_N^{\mathfrak{M}_Q}(F_0,\ldots,F_{K-1})\right| \ll_{\Delta,d,\sigma} R^{(K-1)d-2L}\sum_{q\le2Q^L}q^{-1-\eta}\prod_{i=0}^{K-1}\|\mu_q*F_i\|_{\ell^K(\mathbb{Z}^d)}
+Q^{-2}R^{Kd/2-2L}\prod_{i=0}^{K-1}\|F_i\|_{\ell^2(\mathbb{Z}^d)}. \tag*{(58)} Λ N M Q ( F 0 , … , F K − 1 ) ≪ Δ , d , σ R ( K − 1 ) d − 2 L q ≤ 2 Q L ∑ q − 1 − η i = 0 ∏ K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) + Q − 2 R K d /2 − 2 L i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (58) Reduction to the major arc form
For q ≥ 1 q\ge1 q ≥ 1 a ∈ ( Z / q Z ) L a\in(\mathbb{Z}/q\mathbb{Z})^L a ∈ ( Z / q Z ) L θ ∈ T L \theta\in\mathbb{T}^L θ ∈ T L G i : Z d × ( Z / q Z ) d → C G_i:\mathbb{Z}^d\times(\mathbb{Z}/q\mathbb{Z})^d\to\mathbb{C} G i : Z d × ( Z / q Z ) d → C
M q , a , θ ( G 0 , … , G K − 1 ) : = E r m o d q ∑ u ∈ ( Z d ) K e q ( Q a ( r ) ) e ( Q θ ( u ) ) ∏ i = 0 K − 1 G i ( u i , r i ) . (59) \mathcal{M}_{q,a,\theta}(G_0,\ldots,G_{K-1}):=\mathbb{E}_{r\mathbin{\mathrm{mod}}q}\sum_{u\in(\mathbb{Z}^d)^K}e_q(Q_a(r))e(Q_\theta(u))\prod_{i=0}^{K-1}G_i(u_i,r_i). \tag*{(59)} M q , a , θ ( G 0 , … , G K − 1 ) := E r mod q u ∈ ( Z d ) K ∑ e q ( Q a ( r )) e ( Q θ ( u )) i = 0 ∏ K − 1 G i ( u i , r i ) . (59) Here r = ( r 0 , … , r K − 1 ) r=(r_0,\ldots,r_{K-1}) r = ( r 0 , … , r K − 1 ) ( Z / q Z ) d K (\mathbb{Z}/q\mathbb{Z})^{dK} ( Z / q Z ) d K u = ( u 0 , … , u K − 1 ) u=(u_0,\ldots,u_{K-1}) u = ( u 0 , … , u K − 1 ) ( Z d ) K (\mathbb{Z}^d)^K ( Z d ) K ( u i , r i ) (u_i,r_i) ( u i , r i ) G i G_i G i u i u_i u i r i r_i r i
For every ( u , r ) ∈ Z d × ( Z / q Z ) d (u,r)\in\mathbb{Z}^d\times(\mathbb{Z}/q\mathbb{Z})^d ( u , r ) ∈ Z d × ( Z / q Z ) d y ∈ Z d y\in\mathbb{Z}^d y ∈ Z d
y ≡ r ( m o d q ) , u − y ∈ { 0 , … , q − 1 } d . y\equiv r\pmod q,\qquad u-y\in\{0,\ldots,q-1\}^d. y ≡ r ( mod q ) , u − y ∈ { 0 , … , q − 1 } d . For an input F i : Z d → C F_i:\mathbb{Z}^d\to\mathbb{C} F i : Z d → C G i , q : Z d × ( Z / q Z ) d → C G_{i,q}:\mathbb{Z}^d\times(\mathbb{Z}/q\mathbb{Z})^d\to\mathbb{C} G i , q : Z d × ( Z / q Z ) d → C
G i , q ( u , r ) : = ( μ q ∗ F i ) ( y ) , (60) G_{i,q}(u,r):=(\mu_q*F_i)(y), \tag*{(60)} G i , q ( u , r ) := ( μ q ∗ F i ) ( y ) , (60) where y y y r r r q q q μ q ∗ F i \mu_q*F_i μ q ∗ F i
Lemma 8.2. Suppose 1 ≤ q ≤ 2 Q L 1\le q\le2Q^L 1 ≤ q ≤ 2 Q L a ∈ ( Z / q Z ) L a\in(\mathbb{Z}/q\mathbb{Z})^L a ∈ ( Z / q Z ) L ∥ θ ∥ ∞ ≤ Q / R 2 \|\theta\|_\infty\le Q/R^2 ∥ θ ∥ ∞ ≤ Q / R 2 Then , for all functions F 0 , … , F K − 1 : Z d → C F_0,\ldots,F_{K-1}:\mathbb{Z}^d\to\mathbb{C} F 0 , … , F K − 1 : Z d → C [ N ] d [N]^d [ N ] d
∣ S a / q + θ ( F 0 , … , F K − 1 ) − M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ ≪ K , d , σ Q − L ( L + 2 ) − 2 N K d / 2 ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (61) \left|S_{a/q+\theta}(F_0,\ldots,F_{K-1})-\mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})\right|
\ll_{K,d,\sigma}Q^{-L(L+2)-2}N^{Kd/2}\prod_{i=0}^{K-1}\|F_i\|_{\ell^2(\mathbb{Z}^d)}. \tag*{(61)} S a / q + θ ( F 0 , … , F K − 1 ) − M q , a , θ ( G 0 , q , … , G K − 1 , q ) ≪ K , d , σ Q − L ( L + 2 ) − 2 N K d /2 i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (61) Proof. Since every ∣ z i − z j ∣ 2 |z_i-z_j|^2 ∣ z i − z j ∣ 2
e ( Q a / q + θ ( z ) ) = e q ( Q a ( z ) ) e ( Q θ ( z ) ) . e(Q_{a/q+\theta}(z))=e_q(Q_a(z))e(Q_\theta(z)). e ( Q a / q + θ ( z )) = e q ( Q a ( z )) e ( Q θ ( z )) . Consequently,
S a / q + θ ( F 0 , … , F K − 1 ) = ∑ z ∈ ( Z d ) K e q ( Q a ( z ) ) e ( Q θ ( z ) ) ∏ i = 0 K − 1 F i ( z i ) . (62) S_{a/q+\theta}(F_0,\ldots,F_{K-1})=\sum_{z\in(\mathbb{Z}^d)^K}e_q(Q_a(z))e(Q_\theta(z))\prod_{i=0}^{K-1}F_i(z_i). \tag*{(62)} S a / q + θ ( F 0 , … , F K − 1 ) = z ∈ ( Z d ) K ∑ e q ( Q a ( z )) e ( Q θ ( z )) i = 0 ∏ K − 1 F i ( z i ) . (62) We now rewrite the major arc form in the variables appearing in (62) . Fix r = ( r 0 , … , r K − 1 ) r=(r_0,\ldots,r_{K-1}) r = ( r 0 , … , r K − 1 ) 0 ≤ i ≤ K − 1 0\le i\le K-1 0 ≤ i ≤ K − 1 u i ∈ Z d u_i\in\mathbb{Z}^d u i ∈ Z d
u i = y i + s i , y i ≡ r i ( m o d q ) , s i ∈ { 0 , … , q − 1 } d . u_i=y_i+s_i,\qquad y_i\equiv r_i\pmod q,\qquad s_i\in\{0,\ldots,q-1\}^d. u i = y i + s i , y i ≡ r i ( mod q ) , s i ∈ { 0 , … , q − 1 } d . It follows from (60) that
M q , a , θ ( G 0 , q , … , G K − 1 , q ) = E r m o d q ∑ y ∈ ( Z d ) K y i ≡ r i ( m o d q ) ( 0 ≤ i ≤ K − 1 ) e q ( Q a ( r ) ) ∏ i = 0 K − 1 ( μ q ∗ F i ) ( y i ) ∑ s i ∈ { 0 , … , q − 1 } d 0 ≤ i ≤ K − 1 e ( Q θ ( y + s ) ) . \mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})=\mathbb{E}_{r\ \mathrm{mod}\ q}\sum_{\substack{y\in(\mathbb{Z}^{d})^{K}\\ y_i\equiv r_i\pmod q\ (0\leq i\leq K-1)}}e_q(\mathcal{Q}_a(r))\prod_{i=0}^{K-1}(\mu_q*F_i)(y_i)\sum_{\substack{s_i\in\{0,\ldots,q-1\}^{d}\\0\leq i\leq K-1}}e(\mathcal{Q}_\theta(y+s)). M q , a , θ ( G 0 , q , … , G K − 1 , q ) = E r mod q y ∈ ( Z d ) K y i ≡ r i ( mod q ) ( 0 ≤ i ≤ K − 1 ) ∑ e q ( Q a ( r )) i = 0 ∏ K − 1 ( μ q ∗ F i ) ( y i ) s i ∈ { 0 , … , q − 1 } d 0 ≤ i ≤ K − 1 ∑ e ( Q θ ( y + s )) . Expanding the convolutions gives
M q , a , θ ( G 0 , q , … , G K − 1 , q ) = 1 N q K d E r m o d q ∑ h i ∈ { 0 , … , N q − 1 } d 0 ≤ i ≤ K − 1 ∑ y ∈ ( Z d ) K y i ≡ r i ( m o d q ) ( 0 ≤ i ≤ K − 1 ) e q ( Q a ( r ) ) ∏ i = 0 K − 1 F i ( y i − q h i ) ∑ s i ∈ { 0 , … , q − 1 } d 0 ≤ i ≤ K − 1 e ( Q θ ( y + s ) ) . \mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})=\frac{1}{N_q^{Kd}}\mathbb{E}_{r\ \mathrm{mod}\ q}\sum_{\substack{h_i\in\{0,\ldots,N_q-1\}^{d}\\0\leq i\leq K-1}}\sum_{\substack{y\in(\mathbb{Z}^{d})^{K}\\y_i\equiv r_i\pmod q\ (0\leq i\leq K-1)}}e_q(\mathcal{Q}_a(r))\prod_{i=0}^{K-1}F_i(y_i-qh_i)\sum_{\substack{s_i\in\{0,\ldots,q-1\}^{d}\\0\leq i\leq K-1}}e(\mathcal{Q}_\theta(y+s)). M q , a , θ ( G 0 , q , … , G K − 1 , q ) = N q K d 1 E r mod q h i ∈ { 0 , … , N q − 1 } d 0 ≤ i ≤ K − 1 ∑ y ∈ ( Z d ) K y i ≡ r i ( mod q ) ( 0 ≤ i ≤ K − 1 ) ∑ e q ( Q a ( r )) i = 0 ∏ K − 1 F i ( y i − q h i ) s i ∈ { 0 , … , q − 1 } d 0 ≤ i ≤ K − 1 ∑ e ( Q θ ( y + s )) . For fixed h = ( h 0 , … , h K − 1 ) h=(h_0,\ldots,h_{K-1}) h = ( h 0 , … , h K − 1 )
z i : = y i − q h i ( 0 ≤ i ≤ K − 1 ) . z_i:=y_i-qh_i\qquad(0\leq i\leq K-1). z i := y i − q h i ( 0 ≤ i ≤ K − 1 ) . Then z i ≡ r i ( m o d q ) z_i\equiv r_i\pmod q z i ≡ r i ( mod q ) y + s = z + q h + s y+s=z+qh+s y + s = z + q h + s e q ( Q a ( r ) ) = e q ( Q a ( z ) ) e_q(\mathcal{Q}_a(r))=e_q(\mathcal{Q}_a(z)) e q ( Q a ( r )) = e q ( Q a ( z )) z ∈ ( Z d ) K z\in(\mathbb{Z}^{d})^{K} z ∈ ( Z d ) K r ( m o d q ) r\pmod q r ( mod q ) z i ≡ r i ( m o d q ) z_i\equiv r_i\pmod q z i ≡ r i ( mod q ) i i i r r r q − K d q^{-Kd} q − K d
M q , a , θ ( G 0 , q , … , G K − 1 , q ) = 1 ( q N q ) K d ∑ z ∈ ( Z d ) K e q ( Q a ( z ) ) ∏ i = 0 K − 1 F i ( z i ) ∑ h i ∈ { 0 , … , N q − 1 } d s i ∈ { 0 , … , q − 1 } d 0 ≤ i ≤ K − 1 e ( Q θ ( z + q h + s ) ) . \mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})=\frac{1}{(qN_q)^{Kd}}\sum_{z\in(\mathbb{Z}^{d})^{K}}e_q(\mathcal{Q}_a(z))\prod_{i=0}^{K-1}F_i(z_i)\sum_{\substack{h_i\in\{0,\ldots,N_q-1\}^{d}\\s_i\in\{0,\ldots,q-1\}^{d}\\0\leq i\leq K-1}}e(\mathcal{Q}_\theta(z+qh+s)). M q , a , θ ( G 0 , q , … , G K − 1 , q ) = ( q N q ) K d 1 z ∈ ( Z d ) K ∑ e q ( Q a ( z )) i = 0 ∏ K − 1 F i ( z i ) h i ∈ { 0 , … , N q − 1 } d s i ∈ { 0 , … , q − 1 } d 0 ≤ i ≤ K − 1 ∑ e ( Q θ ( z + q h + s )) . Finally, for each i i i ( h i , s i ) ↦ t i : = q h i + s i (h_i,s_i)\mapsto t_i:=qh_i+s_i ( h i , s i ) ↦ t i := q h i + s i
{ 0 , … , N q − 1 } d × { 0 , … , q − 1 } d onto { 0 , … , q N q − 1 } d . \{0,\ldots,N_q-1\}^{d}\times\{0,\ldots,q-1\}^{d}\quad\text{onto}\quad\{0,\ldots,qN_q-1\}^{d}. { 0 , … , N q − 1 } d × { 0 , … , q − 1 } d onto { 0 , … , q N q − 1 } d . Consequently,
M q , a , θ ( G 0 , q , … , G K − 1 , q ) = ∑ z ∈ ( Z d ) K e q ( Q a ( z ) ) ∏ i = 0 K − 1 F i ( z i ) × 1 ( q N q ) K d ∑ t i ∈ { 0 , … , q N q − 1 } d 0 ≤ i ≤ K − 1 e ( Q θ ( z + t ) ) . (63) \mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})=\sum_{z\in(\mathbb{Z}^{d})^{K}}e_q(\mathcal{Q}_a(z))\prod_{i=0}^{K-1}F_i(z_i)\times\frac{1}{(qN_q)^{Kd}}\sum_{\substack{t_i\in\{0,\ldots,qN_q-1\}^{d}\\0\leq i\leq K-1}}e(\mathcal{Q}_\theta(z+t)). \tag*{(63)} M q , a , θ ( G 0 , q , … , G K − 1 , q ) = z ∈ ( Z d ) K ∑ e q ( Q a ( z )) i = 0 ∏ K − 1 F i ( z i ) × ( q N q ) K d 1 t i ∈ { 0 , … , q N q − 1 } d 0 ≤ i ≤ K − 1 ∑ e ( Q θ ( z + t )) . (63) Comparing (62) and (63) , the rational phase and the values of the F i F_i F i e ( Q θ ( z ) ) e(\mathcal{Q}_\theta(z)) e ( Q θ ( z )) e ( Q θ ( z + t ) ) e(\mathcal{Q}_\theta(z+t)) e ( Q θ ( z + t )) t t t
For 0 ≤ i ≤ K − 1 0\leq i\leq K-1 0 ≤ i ≤ K − 1
∇ x i Q θ ( x ) = 2 ∑ j : { i , j } ∈ E θ i j ( x i − x j ) . \nabla_{x_i}\mathcal{Q}_\theta(x)=2\sum_{j:\{i,j\}\in E}\theta_{ij}(x_i-x_j). ∇ x i Q θ ( x ) = 2 j : { i , j } ∈ E ∑ θ ij ( x i − x j ) . If z i ∈ [ N ] d z_i\in[N]^d z i ∈ [ N ] d t i ∈ { 0 , … , q N q − 1 } d t_i\in\{0,\ldots,qN_q-1\}^{d} t i ∈ { 0 , … , q N q − 1 } d i i i x x x z z z z + t z+t z + t
max i , j ∣ x i − x j ∣ ≪ d N + q N q ≪ d N . \max_{i,j}|x_i-x_j|\ll_d N+qN_q\ll_d N. i , j max ∣ x i − x j ∣ ≪ d N + q N q ≪ d N . Since ∥ θ ∥ ∞ ≤ Q / R 2 \lVert\theta\rVert_\infty\le Q/R^2 ∥ θ ∥ ∞ ≤ Q / R 2 q N q ≤ R Q − L ( L + 2 ) − 3 qN_q \le RQ^{-L(L+2)-3} q N q ≤ R Q − L ( L + 2 ) − 3
∣ e ( Q θ ( z ) ) − e ( Q θ ( z + t ) ) ∣ ≪ K , d Q N R 2 q N q ≪ K , d , σ Q − L ( L + 2 ) − 2 . \left|e(Q_\theta(z))-e(Q_\theta(z+t))\right| \ll_{K,d} \frac{QN}{R^2}qN_q \ll_{K,d,\sigma} Q^{-L(L+2)-2}. ∣ e ( Q θ ( z )) − e ( Q θ ( z + t )) ∣ ≪ K , d R 2 QN q N q ≪ K , d , σ Q − L ( L + 2 ) − 2 . Here we used R = ⌊ σ N ⌋ R=\lfloor\sigma N\rfloor R = ⌊ σ N ⌋ (56) , which imply N ≪ σ R N\ll_\sigma R N ≪ σ R t t t
∣ S a / q + θ ( F 0 , … , F K − 1 ) − M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ ≪ K , d , σ Q − L ( L + 2 ) − 2 ∏ i = 0 K − 1 ∑ z i ∈ [ N ] d ∣ F i ( z i ) ∣ . (64) \left|S_{a/q+\theta}(F_0,\ldots,F_{K-1})-M_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})\right|
\ll_{K,d,\sigma} Q^{-L(L+2)-2}\prod_{i=0}^{K-1}\sum_{z_i\in[N]^d}|F_i(z_i)|. \tag*{(64)} S a / q + θ ( F 0 , … , F K − 1 ) − M q , a , θ ( G 0 , q , … , G K − 1 , q ) ≪ K , d , σ Q − L ( L + 2 ) − 2 i = 0 ∏ K − 1 z i ∈ [ N ] d ∑ ∣ F i ( z i ) ∣. (64) Cauchy–Schwarz in each factor proves (61) .
The complete sum
With normalized counting measure, define the complete cut operator
K q , a , d S , T : L 2 ( ( ( Z / q Z ) d ) T ) ⟶ L 2 ( ( ( Z / q Z ) d ) S ) , \mathcal{K}_{q,a,d}^{S,T}:L^2\left(((\mathbb{Z}/q\mathbb{Z})^d)^T\right)\longrightarrow L^2\left(((\mathbb{Z}/q\mathbb{Z})^d)^S\right), K q , a , d S , T : L 2 ( (( Z / q Z ) d ) T ) ⟶ L 2 ( (( Z / q Z ) d ) S ) , ( K q , a , d S , T H ) ( r S ) : = E r T mod q e q ( − 2 ∑ i ∈ S , j ∈ T a i j r i ⋅ r j ) H ( r T ) . \left(\mathcal{K}_{q,a,d}^{S,T}H\right)(r_S):=\mathbb{E}_{r_T\mathbin{\operatorname{mod}}q}e_q\left(-2\sum_{i\in S,j\in T}a_{ij}r_i\cdot r_j\right)H(r_T). ( K q , a , d S , T H ) ( r S ) := E r T mod q e q − 2 i ∈ S , j ∈ T ∑ a ij r i ⋅ r j H ( r T ) . We now compute the 2 → 2 2\to2 2 → 2
Lemma 8.3. One has
∥ K q , a , d S , T ∥ 2 → 2 = ∣ im ( 2 a S , T ) ∣ − d / 2 . (65) \left\|\mathcal{K}_{q,a,d}^{S,T}\right\|_{2\to2}=\left|\operatorname{im}(2a_{S,T})\right|^{-d/2}. \tag*{(65)} K q , a , d S , T 2 → 2 = ∣ im ( 2 a S , T ) ∣ − d /2 . (65) Proof. By the same argument as in the proof of Lemma 6.2,
∥ K q , a , d S , T ∥ 2 → 2 = ∥ K q , a , 1 S , T ∥ 2 → 2 . (66) \left\|\mathcal{K}_{q,a,d}^{S,T}\right\|_{2\to2}=\left\|\mathcal{K}_{q,a,1}^{S,T}\right\|_{2\to2}. \tag*{(66)} K q , a , d S , T 2 → 2 = K q , a , 1 S , T 2 → 2 . (66) It therefore remains to compute ∥ K q , a , 1 S , T ∥ 2 → 2 \left\|\mathcal{K}_{q,a,1}^{S,T}\right\|_{2\to2} K q , a , 1 S , T 2 → 2
Write K : = K q , a , 1 S , T \mathcal{K}:=\mathcal{K}_{q,a,1}^{S,T} K := K q , a , 1 S , T f f f ( Z / q Z ) S (\mathbb{Z}/q\mathbb{Z})^S ( Z / q Z ) S K K ∗ \mathcal{K}\mathcal{K}^* K K ∗
( K K ∗ f ) ( r S ) = E r S ′ mod q f ( r S ′ ) E r T mod q e q ( ( r S ′ − r S ) T 2 a S , T r T ) = q − ∣ S ∣ ∑ r S ′ mod q ( 2 a S , T ) T ( r S ′ − r S ) = 0 f ( r S ′ ) , \begin{aligned}
(\mathcal{K}\mathcal{K}^*f)(r_S)=\mathbb{E}_{r'_S\mathbin{\operatorname{mod}}q}f(r'_S)\mathbb{E}_{r_T\mathbin{\operatorname{mod}}q}e_q\left((r'_S-r_S)^{\mathsf T}2a_{S,T}r_T\right) \\
&=q^{-|S|}\sum_{\substack{r'_S\mathbin{\operatorname{mod}}q\\(2a_{S,T})^{\mathsf T}(r'_S-r_S)=0}}f(r'_S),
\end{aligned} ( K K ∗ f ) ( r S ) = E r S ′ mod q f ( r S ′ ) E r T mod q e q ( ( r S ′ − r S ) T 2 a S , T r T ) = q − ∣ S ∣ r S ′ mod q ( 2 a S , T ) T ( r S ′ − r S ) = 0 ∑ f ( r S ′ ) , where the second equality follows from the orthogonality relation. Making the substitution h = r S − r S ′ h=r_S-r'_S h = r S − r S ′
( K K ∗ f ) ( r S ) = q − ∣ S ∣ ∑ h ∈ ker ( ( 2 a S , T ) T ) f ( r S − h ) . (\mathcal{K}\mathcal{K}^*f)(r_S)=q^{-|S|}\sum_{h\in\ker((2a_{S,T})^{\mathsf T})}f(r_S-h). ( K K ∗ f ) ( r S ) = q − ∣ S ∣ h ∈ k e r (( 2 a S , T ) T ) ∑ f ( r S − h ) . Hence, by Cauchy–Schwarz,
∥ K K ∗ f ∥ 2 2 = q − 2 ∣ S ∣ E r S mod q ∣ ∑ h ∈ ker ( ( 2 a S , T ) T ) f ( r S − h ) ∣ 2 ≤ q − 2 ∣ S ∣ ∣ ker ( ( 2 a S , T ) T ) ∣ ∑ h ∈ ker ( ( 2 a S , T ) T ) E r S mod q ∣ f ( r S − h ) ∣ 2 = ( ∣ ker ( ( 2 a S , T ) T ) ∣ q ∣ S ∣ ) 2 ∥ f ∥ 2 2 . \begin{aligned}
\|\mathcal{K}\mathcal{K}^*f\|_2^2
=q^{-2|S|}\mathbb{E}_{r_S\mathbin{\operatorname{mod}}q}\left|\sum_{h\in\ker((2a_{S,T})^{\mathsf T})}f(r_S-h)\right|^2 \\
&\le q^{-2|S|}|\ker((2a_{S,T})^{\mathsf T})|
\sum_{h\in\ker((2a_{S,T})^{\mathsf T})}\mathbb{E}_{r_S\mathbin{\operatorname{mod}}q}|f(r_S-h)|^2 \\
&=\left(\frac{|\ker((2a_{S,T})^{\mathsf T})|}{q^{|S|}}\right)^2\|f\|_2^2.
\end{aligned} ∥ K K ∗ f ∥ 2 2 = q − 2∣ S ∣ E r S mod q h ∈ k e r (( 2 a S , T ) T ) ∑ f ( r S − h ) 2 ≤ q − 2∣ S ∣ ∣ ker (( 2 a S , T ) T ) ∣ h ∈ k e r (( 2 a S , T ) T ) ∑ E r S mod q ∣ f ( r S − h ) ∣ 2 = ( q ∣ S ∣ ∣ ker (( 2 a S , T ) T ) ∣ ) 2 ∥ f ∥ 2 2 . This bound is attained by f = 1 f=1 f = 1 K K ∗ 1 = ( ∣ ker ( ( 2 a S , T ) T ) ∣ / q ∣ S ∣ ) 1 \mathcal{K}\mathcal{K}^{*}1=(|\ker((2a_{S,T})^{\mathsf T})|/q^{|S|})1 K K ∗ 1 = ( ∣ ker (( 2 a S , T ) T ) ∣/ q ∣ S ∣ ) 1
∥ K K ∗ ∥ 2 → 2 = ∣ ker ( ( 2 a S , T ) T ) ∣ q ∣ S ∣ . \|\mathcal{K}\mathcal{K}^{*}\|_{2\to2}=\frac{|\ker((2a_{S,T})^{\mathsf T})|}{q^{|S|}}. ∥ K K ∗ ∥ 2 → 2 = q ∣ S ∣ ∣ ker (( 2 a S , T ) T ) ∣ . Finally, the orthogonality relation gives
∣ ker ( ( 2 a S , T ) T ) ∣ = ∑ r S m o d q E r T m o d q e q ( r S T 2 a S , T r T ) = q ∣ S ∣ − ∣ T ∣ ∣ ker ( 2 a S , T ) ∣ = q ∣ S ∣ ∣ im ( 2 a S , T ) ∣ . \begin{aligned}
|\ker((2a_{S,T})^{\mathsf T})|=\sum_{r_S\bmod q}\mathbb{E}_{r_T\bmod q}e_q\left(r_S^{\mathsf T}2a_{S,T}r_T\right) \\
&=q^{|S|-|T|}|\ker(2a_{S,T})| \\
&=\frac{q^{|S|}}{|\operatorname{im}(2a_{S,T})|}.
\end{aligned} ∣ ker (( 2 a S , T ) T ) ∣ = r S mod q ∑ E r T mod q e q ( r S T 2 a S , T r T ) = q ∣ S ∣ − ∣ T ∣ ∣ ker ( 2 a S , T ) ∣ = ∣ im ( 2 a S , T ) ∣ q ∣ S ∣ . where the last equality follows from
q ∣ T ∣ = ∣ ker ( 2 a S , T ) ∣ ∣ im ( 2 a S , T ) ∣ . q^{|T|}=|\ker(2a_{S,T})||\operatorname{im}(2a_{S,T})|. q ∣ T ∣ = ∣ ker ( 2 a S , T ) ∣∣ im ( 2 a S , T ) ∣. Consequently,
∥ K q , a , 1 S , T ∥ 2 → 2 2 = ∥ K K ∗ ∥ 2 → 2 = ∣ im ( 2 a S , T ) ∣ − 1 . (67) \|\mathcal{K}_{q,a,1}^{S,T}\|_{2\to2}^{2}=\|\mathcal{K}\mathcal{K}^{*}\|_{2\to2}=|\operatorname{im}(2a_{S,T})|^{-1}. \tag*{(67)} ∥ K q , a , 1 S , T ∥ 2 → 2 2 = ∥ K K ∗ ∥ 2 → 2 = ∣ im ( 2 a S , T ) ∣ − 1 . (67) Combining (66) and (67) proves (65) .
We need an unrestricted matrix moment and a stronger moment when one entry is a unit.
Lemma 8.4 (Matrix moments). Let p p p m m m s s s t t t τ > s + t − 1 \tau>s+t-1 τ > s + t − 1 M ∈ Mat s × t ( Z / p m Z ) \mathbf{M}\in\operatorname{Mat}_{s\times t}(\mathbb{Z}/p^{m}\mathbb{Z}) M ∈ Mat s × t ( Z / p m Z ) 1 ≤ i ≤ s 1\leq i\leq s 1 ≤ i ≤ s 1 ≤ j ≤ t 1\leq j\leq t 1 ≤ j ≤ t
E M ∣ im ( 2 M ) ∣ − τ ≪ s , t , τ p − m s t , (68) \mathbb{E}_{\mathbf{M}}|\operatorname{im}(2\mathbf{M})|^{-\tau}\ll_{s,t,\tau}p^{-mst}, \tag*{(68)} E M ∣ im ( 2 M ) ∣ − τ ≪ s , t , τ p − m s t , (68) E M [ 1 M i j ∈ ( Z / p m Z ) × ∣ im ( 2 M ) ∣ − τ ] ≪ s , t , τ p − m [ τ + ( s − 1 ) ( t − 1 ) ] . (69) \mathbb{E}_{\mathbf{M}}\left[1_{M_{ij}\in(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}}|\operatorname{im}(2\mathbf{M})|^{-\tau}\right]\ll_{s,t,\tau}p^{-m[\tau+(s-1)(t-1)]}. \tag*{(69)} E M [ 1 M ij ∈ ( Z / p m Z ) × ∣ im ( 2 M ) ∣ − τ ] ≪ s , t , τ p − m [ τ + ( s − 1 ) ( t − 1 )] . (69) The implicit constants depend only on s s s t t t τ \tau τ
Proof. Put
W s , t ( p m ) : = ∑ M ∈ Mat s × t ( Z / p m Z ) ∣ im M ∣ − τ , W_{s,t}(p^{m}):=\sum_{\mathbf{M}\in\operatorname{Mat}_{s\times t}(\mathbb{Z}/p^{m}\mathbb{Z})}|\operatorname{im}\mathbf{M}|^{-\tau}, W s , t ( p m ) := M ∈ Mat s × t ( Z / p m Z ) ∑ ∣ im M ∣ − τ , with the conventions W 0 , t ( p m ) = W s , 0 ( p m ) = W s , t ( 1 ) = 1 W_{0,t}(p^{m})=W_{s,0}(p^{m})=W_{s,t}(1)=1 W 0 , t ( p m ) = W s , 0 ( p m ) = W s , t ( 1 ) = 1
The matrices divisible by p p p W s , t ( p m − 1 ) W_{s,t}(p^{m-1}) W s , t ( p m − 1 )
M = ( u b c D ) , \mathbf{M}=\begin{pmatrix}u&\mathbf{b}\\ \mathbf{c}&D\end{pmatrix}, M = ( u c b D ) , where b \mathbf{b} b c \mathbf{c} c M \mathbf{M} M
( u 0 0 M ′ ) , M ′ = D − c u − 1 b . \begin{pmatrix}u&0\\0&\mathbf{M}'\end{pmatrix},
\qquad\mathbf{M}'=D-\mathbf{c}u^{-1}\mathbf{b}. ( u 0 0 M ′ ) , M ′ = D − c u − 1 b . These operations preserve image cardinality, and hence ∣ im M ∣ = p m ∣ im M ′ ∣ |\operatorname{im}\mathbf{M}|=p^{m}|\operatorname{im}\mathbf{M}'| ∣ im M ∣ = p m ∣ im M ′ ∣ O s , t ( p m ( s + t − 1 ) ) O_{s,t}(p^{m(s+t-1)}) O s , t ( p m ( s + t − 1 ) ) u u u b \mathbf{b} b c \mathbf{c} c C ⟼ M ′ = C − c u − 1 b \mathbf{C}\longmapsto\mathbf{M}'=\mathbf{C}-\mathbf{c}u^{-1}\mathbf{b} C ⟼ M ′ = C − c u − 1 b Mat ( s − 1 ) × ( t − 1 ) ( Z / p m Z ) \operatorname{Mat}_{(s-1)\times(t-1)}(\mathbb{Z}/p^{m}\mathbb{Z}) Mat ( s − 1 ) × ( t − 1 ) ( Z / p m Z ) M ′ \mathbf{M}' M ′
W s , t ( p m ) ≤ W s , t ( p m − 1 ) + c s , t p − m ( τ − s − t + 1 ) W s − 1 , t − 1 ( p m ) . (70) W_{s,t}(p^{m})\leq W_{s,t}(p^{m-1})+c_{s,t}p^{-m(\tau-s-t+1)}W_{s-1,t-1}(p^{m}). \tag*{(70)} W s , t ( p m ) ≤ W s , t ( p m − 1 ) + c s , t p − m ( τ − s − t + 1 ) W s − 1 , t − 1 ( p m ) . (70) We now induct on min ( s , t ) \min(s,t) min ( s , t ) min ( s , t ) = 0 \min(s,t)=0 min ( s , t ) = 0 W s , t ( p m ) = 1 W_{s,t}(p^m)=1 W s , t ( p m ) = 1 s , t ≥ 1 s,t\ge1 s , t ≥ 1 W s − 1 , t − 1 ( p r ) ≪ s , t , τ 1 W_{s-1,t-1}(p^r)\ll_{s,t,\tau}1 W s − 1 , t − 1 ( p r ) ≪ s , t , τ 1 p p p r r r δ : = τ − s − t + 1 > 0 \delta:=\tau-s-t+1>0 δ := τ − s − t + 1 > 0 m m m 0 0 0
W s , t ( p m ) ≤ W s , t ( p m − 1 ) + c s , t p − m δ W s − 1 , t − 1 ( p m ) ≤ W s , t ( p m − 2 ) + c s , t p − ( m − 1 ) δ W s − 1 , t − 1 ( p m − 1 ) + c s , t p − m δ W s − 1 , t − 1 ( p m ) ≤ ⋮ ≤ W s , t ( 1 ) + c s , t ∑ r = 1 m p − r δ W s − 1 , t − 1 ( p r ) ≪ 1 + ∑ r = 1 m p − r δ ≪ 1. \begin{aligned}
W_{s,t}(p^m)&\le W_{s,t}(p^{m-1})+c_{s,t}p^{-m\delta}W_{s-1,t-1}(p^m)\\
&\le W_{s,t}(p^{m-2})+c_{s,t}p^{-(m-1)\delta}W_{s-1,t-1}(p^{m-1})+c_{s,t}p^{-m\delta}W_{s-1,t-1}(p^m)\\
&\phantom{\le}\vdots\\
&\le W_{s,t}(1)+c_{s,t}\sum_{r=1}^{m}p^{-r\delta}W_{s-1,t-1}(p^r)\\
&\ll1+\sum_{r=1}^{m}p^{-r\delta}\ll1.
\end{aligned} W s , t ( p m ) ≤ W s , t ( p m − 1 ) + c s , t p − m δ W s − 1 , t − 1 ( p m ) ≤ W s , t ( p m − 2 ) + c s , t p − ( m − 1 ) δ W s − 1 , t − 1 ( p m − 1 ) + c s , t p − m δ W s − 1 , t − 1 ( p m ) ≤ ⋮ ≤ W s , t ( 1 ) + c s , t r = 1 ∑ m p − r δ W s − 1 , t − 1 ( p r ) ≪ 1 + r = 1 ∑ m p − r δ ≪ 1. Consequently,
E M ∣ im M ∣ − τ = p − m s t W s , t ( p m ) ≪ p − m s t . (71) \mathbb{E}_{M}|\operatorname{im} M|^{-\tau}=p^{-mst}W_{s,t}(p^m)\ll p^{-mst}. \tag*{(71)} E M ∣ im M ∣ − τ = p − m s t W s , t ( p m ) ≪ p − m s t . (71) It remains to pass from M M M 2 M 2M 2 M p p p 2 2 2 p m p^m p m ∣ im ( 2 M ) ∣ = ∣ im M ∣ |\operatorname{im}(2M)|=|\operatorname{im}M| ∣ im ( 2 M ) ∣ = ∣ im M ∣ (68) .
Suppose now that p = 2 p=2 p = 2 m ≥ 2 m\ge2 m ≥ 2 M ‾ \overline{M} M M M M 2 m − 1 2^{m-1} 2 m − 1 ∣ im ( 2 M m o d 2 m ) ∣ = ∣ im M ‾ ∣ |\operatorname{im}(2M\bmod2^m)|=|\operatorname{im}\overline{M}| ∣ im ( 2 M mod 2 m ) ∣ = ∣ im M ∣ 2 m − 1 2^{m-1} 2 m − 1 2 s t 2^{st} 2 s t 2 m 2^m 2 m
E M m o d 2 m ∣ im ( 2 M ) ∣ − τ = E M ‾ m o d 2 m − 1 ∣ im M ‾ ∣ − τ ≪ 2 − ( m − 1 ) s t ≪ 2 − m s t . \mathbb{E}_{M\bmod2^m}|\operatorname{im}(2M)|^{-\tau}=\mathbb{E}_{\overline{M}\bmod2^{m-1}}|\operatorname{im}\overline{M}|^{-\tau}\ll2^{-(m-1)st}\ll2^{-mst}. E M mod 2 m ∣ im ( 2 M ) ∣ − τ = E M mod 2 m − 1 ∣ im M ∣ − τ ≪ 2 − ( m − 1 ) s t ≪ 2 − m s t . The constant lost in the final inequality depends only on s s s t t t τ \tau τ m = 1 m=1 m = 1 (69) follows immediately from the proof of (68) : use the prescribed unit entry M i j M_{ij} M ij p = 2 p=2 p = 2 2 m − 1 2^{m-1} 2 m − 1
For a ∈ ( Z / q Z ) L a\in(\mathbb{Z}/q\mathbb{Z})^L a ∈ ( Z / q Z ) L
s q ( a ) : = ∏ S ∣ T ∈ B K ∣ im ( 2 a S , T ) ∣ − τ w K . (72) \mathfrak{s}_q(a):=\prod_{S\mid T\in\mathcal{B}_K}|\operatorname{im}(2a_{S,T})|^{-\tau w_K}. \tag*{(72)} s q ( a ) := S ∣ T ∈ B K ∏ ∣ im ( 2 a S , T ) ∣ − τ w K . (72) We use the exponent τ : = d ⌊ K 2 / 4 ⌋ / ( 2 L ) \tau:=d\lfloor K^2/4\rfloor/(2L) τ := d ⌊ K 2 /4 ⌋ / ( 2 L ) Proposition 7.1 . Under the hypothesis d ≥ 4 K + 4 d\ge4K+4 d ≥ 4 K + 4 τ > K + 2 \tau>K+2 τ > K + 2 η : = ( τ − K ) / 2 \eta:=(\tau-K)/2 η := ( τ − K ) /2 τ > K + 2 \tau>K+2 τ > K + 2 η > 1 \eta>1 η > 1
Lemma 8.5. If d ≥ 4 K + 4 d\ge4K+4 d ≥ 4 K + 4 η > 1 \eta>1 η > 1 q ≥ 1 q\ge1 q ≥ 1
∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 s q ( a ) ≪ K , d q − 1 − η . (73) \sum_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}\mathfrak{s}_q(a)\ll_{K,d}q^{-1-\eta}. \tag*{(73)} a ∈ ( Z / q Z ) L ( a , q ) = 1 ∑ s q ( a ) ≪ K , d q − 1 − η . (73) Proof. The case q = 1 q=1 q = 1 q = p m q=p^m q = p m e ∈ E e\in\mathcal{E} e ∈ E
I e : = E a ∈ ( Z / p m Z ) L [ 1 a e ∈ ( Z / p m Z ) × s p m ( a ) ] . I_e:=\mathbb{E}_{a\in(\mathbb{Z}/p^m\mathbb{Z})^L}\left[1_{a_e\in(\mathbb{Z}/p^m\mathbb{Z})^\times}\mathfrak{s}_{p^m}(a)\right]. I e := E a ∈ ( Z / p m Z ) L [ 1 a e ∈ ( Z / p m Z ) × s p m ( a ) ] . Fix e ∈ E e\in\mathcal{E} e ∈ E J = ∣ B K ∣ J=|\mathcal{B}_K| J = ∣ B K ∣ 1 , … , J 1,\ldots,J 1 , … , J S ∣ T ∈ B K S\mid T\in\mathcal{B}_K S ∣ T ∈ B K j j j S ∣ T S\mid T S ∣ T E j E_j E j S ∣ T S\mid T S ∣ T c j c_j c j w K w_K w K f S , T f_{S,T} f S , T f j f_j f j
For each balanced cut S ∣ T S \mid T S ∣ T
f S , T ( a S , T ) : = { 1 a e ∈ ( Z / p m Z ) × ∣ im ( 2 a S , T ) ∣ − τ , e crosses S ∣ T , ∣ im ( 2 a S , T ) ∣ − τ , e does not cross S ∣ T . f_{S,T}(a_{S,T}) := \begin{cases} 1_{a_e\in(\mathbb{Z}/p^m\mathbb{Z})^\times}|\operatorname{im}(2a_{S,T})|^{-\tau}, & e\text{ crosses }S\mid T, \\ |\operatorname{im}(2a_{S,T})|^{-\tau}, & e\text{ does not cross }S\mid T. \end{cases} f S , T ( a S , T ) := { 1 a e ∈ ( Z / p m Z ) × ∣ im ( 2 a S , T ) ∣ − τ , ∣ im ( 2 a S , T ) ∣ − τ , e crosses S ∣ T , e does not cross S ∣ T . When e e e S ∣ T S\mid T S ∣ T a e a_e a e a S , T a_{S,T} a S , T E j \mathcal{E}_j E j
For every e ′ ∈ E e' \in\mathcal{E} e ′ ∈ E
∑ S ∣ T ∈ B K e ′ crosses S ∣ T w K = 1 , \sum_{\substack{S\mid T\in\mathcal{B}_K\\ e'\text{ crosses }S\mid T}} w_K = 1, S ∣ T ∈ B K e ′ crosses S ∣ T ∑ w K = 1 , which is exactly the hypothesis of Lemma 6.5. Moreover,
∏ S ∣ T ∈ B K f S , T ( a S , T ) w K = ( ∏ S ∣ T ∈ B K e crosses S ∣ T 1 a e ∈ ( Z / p m Z ) × w K ) ∏ S ∣ T ∈ B K ∣ im ( 2 a S , T ) ∣ − τ w K = 1 a e ∈ ( Z / p m Z ) × s ~ p m ( a ) , \prod_{S\mid T\in\mathcal{B}_K} f_{S,T}(a_{S,T})^{w_K}
=
\left(\prod_{\substack{S\mid T\in\mathcal{B}_K\\ e\text{ crosses }S\mid T}}1_{a_e\in(\mathbb{Z}/p^m\mathbb{Z})^\times}^{w_K}\right)
\prod_{S\mid T\in\mathcal{B}_K}|\operatorname{im}(2a_{S,T})|^{-\tau w_K}
=
1_{a_e\in(\mathbb{Z}/p^m\mathbb{Z})^\times}\widetilde{s}_{p^m}(a), S ∣ T ∈ B K ∏ f S , T ( a S , T ) w K = S ∣ T ∈ B K e crosses S ∣ T ∏ 1 a e ∈ ( Z / p m Z ) × w K S ∣ T ∈ B K ∏ ∣ im ( 2 a S , T ) ∣ − τ w K = 1 a e ∈ ( Z / p m Z ) × s p m ( a ) , where the last equality follows from (43) . Hence
I e = E a ∈ ( Z / p m Z ) L ∏ S ∣ T ∈ B K f S , T ( a S , T ) w K . I_e=\mathbb{E}_{a\in(\mathbb{Z}/p^m\mathbb{Z})^L}\prod_{S\mid T\in\mathcal{B}_K}f_{S,T}(a_{S,T})^{w_K}. I e = E a ∈ ( Z / p m Z ) L S ∣ T ∈ B K ∏ f S , T ( a S , T ) w K . Lemma 6.5 therefore gives
I e ≤ ∏ S ∣ T ∈ B K e crosses S ∣ T ( E a S , T [ 1 a e ∈ ( Z / p m Z ) × ∣ im ( 2 a S , T ) ∣ − τ ] ) w K × ∏ S ∣ T ∈ B K e does not cross S ∣ T ( E a S , T ∣ im ( 2 a S , T ) ∣ − τ ) w K , I_e\leq
\prod_{\substack{S\mid T\in\mathcal{B}_K\\ e\text{ crosses }S\mid T}}
\left(\mathbb{E}_{a_{S,T}}\left[1_{a_e\in(\mathbb{Z}/p^m\mathbb{Z})^\times}|\operatorname{im}(2a_{S,T})|^{-\tau}\right]\right)^{w_K}
\times
\prod_{\substack{S\mid T\in\mathcal{B}_K\\ e\text{ does not cross }S\mid T}}
\left(\mathbb{E}_{a_{S,T}}|\operatorname{im}(2a_{S,T})|^{-\tau}\right)^{w_K}, I e ≤ S ∣ T ∈ B K e crosses S ∣ T ∏ ( E a S , T [ 1 a e ∈ ( Z / p m Z ) × ∣ im ( 2 a S , T ) ∣ − τ ] ) w K × S ∣ T ∈ B K e does not cross S ∣ T ∏ ( E a S , T ∣ im ( 2 a S , T ) ∣ − τ ) w K , where each average in a S , T a_{S,T} a S , T Mat ∣ S ∣ × ∣ T ∣ ( Z / p m Z ) \operatorname{Mat}_{|S|\times|T|}(\mathbb{Z}/p^m\mathbb{Z}) Mat ∣ S ∣ × ∣ T ∣ ( Z / p m Z )
We now estimate the averages on the right. Fix a balanced cut S ∣ T S\mid T S ∣ T e e e S ∣ T S\mid T S ∣ T (69) gives
E a S , T [ 1 a e ∈ ( Z / p m Z ) × ∣ im ( 2 a S , T ) ∣ − τ ] ≪ K , d p − m [ τ + ( ∣ S ∣ − 1 ) ( ∣ T ∣ − 1 ) ] = p − m ( ∣ S ∣ ∣ T ∣ + 1 + 2 η ) , \mathbb{E}_{a_{S,T}}\left[1_{a_e\in(\mathbb{Z}/p^m\mathbb{Z})^\times}|\operatorname{im}(2a_{S,T})|^{-\tau}\right]\ll_{K,d}p^{-m[\tau+(|S|-1)(|T|-1)]}=p^{-m(|S||T|+1+2\eta)}, E a S , T [ 1 a e ∈ ( Z / p m Z ) × ∣ im ( 2 a S , T ) ∣ − τ ] ≪ K , d p − m [ τ + ( ∣ S ∣ − 1 ) ( ∣ T ∣ − 1 )] = p − m ( ∣ S ∣∣ T ∣ + 1 + 2 η ) , where we used ∣ S ∣ + ∣ T ∣ = K |S|+|T|=K ∣ S ∣ + ∣ T ∣ = K 2 η = τ − K 2\eta=\tau-K 2 η = τ − K e e e S ∣ T S\mid T S ∣ T (68) gives
E a S , T ∣ im ( 2 a S , T ) ∣ − τ ≪ K , d p − m ∣ S ∣ ∣ T ∣ . \mathbb{E}_{a_{S,T}}|\operatorname{im}(2a_{S,T})|^{-\tau}\ll_{K,d}p^{-m|S||T|}. E a S , T ∣ im ( 2 a S , T ) ∣ − τ ≪ K , d p − m ∣ S ∣∣ T ∣ . Consequently,
I e ≪ K , d p − m w K ∑ S ∣ T ∈ B K ∣ S ∣ ∣ T ∣ p − m ( 1 + 2 η ) w K # { S ∣ T ∈ B K : e crosses S ∣ T } . I_e\ll_{K,d}p^{-mw_K\sum_{S\mid T\in\mathcal{B}_K}|S||T|}p^{-m(1+2\eta)w_K\#\{S\mid T\in\mathcal{B}_K:e\text{ crosses }S\mid T\}}. I e ≪ K , d p − m w K ∑ S ∣ T ∈ B K ∣ S ∣∣ T ∣ p − m ( 1 + 2 η ) w K # { S ∣ T ∈ B K : e crosses S ∣ T } . By (55) and (43) , I e ≪ K , d p − m ( L + 1 + 2 η ) I_e\ll_{K,d}p^{-m(L+1+2\eta)} I e ≪ K , d p − m ( L + 1 + 2 η )
If ( a , p m ) = 1 (a,p^m)=1 ( a , p m ) = 1 a e a_e a e e ∈ E e\in\mathcal{E} e ∈ E
1 ( a , p m ) = 1 ≤ ∑ e ∈ E 1 a e ∈ ( Z / p m Z ) × , 1_{(a,p^m)=1}\leq\sum_{e\in\mathcal{E}}1_{a_e\in(\mathbb{Z}/p^m\mathbb{Z})^\times}, 1 ( a , p m ) = 1 ≤ e ∈ E ∑ 1 a e ∈ ( Z / p m Z ) × , and hence
∑ a ∈ ( Z / p m Z ) L ( a , p m ) = 1 s p m ( a ) = p m L E a ∈ ( Z / p m Z ) L [ 1 ( a , p m ) = 1 s p m ( a ) ] ≤ p m L ∑ e ∈ E I e ≪ K , d p − m ( 1 + 2 η ) . \sum_{\substack{a\in(\mathbb{Z}/p^m\mathbb{Z})^L\\(a,p^m)=1}} s_{p^m}(a)=p^{mL}\mathbb{E}_{a\in(\mathbb{Z}/p^m\mathbb{Z})^L}\left[\mathbf{1}_{(a,p^m)=1}s_{p^m}(a)\right]\leq p^{mL}\sum_{e\in\mathcal{E}}I_e\ll_{K,d}p^{-m(1+2\eta)}. a ∈ ( Z / p m Z ) L ( a , p m ) = 1 ∑ s p m ( a ) = p m L E a ∈ ( Z / p m Z ) L [ 1 ( a , p m ) = 1 s p m ( a ) ] ≤ p m L e ∈ E ∑ I e ≪ K , d p − m ( 1 + 2 η ) . Now write q = ∏ r = 1 u p r m r q=\prod_{r=1}^{u}p_r^{m_r} q = ∏ r = 1 u p r m r ( Z / q Z ) L ≅ ∏ r = 1 u ( Z / p r m r Z ) L (\mathbb{Z}/q\mathbb{Z})^L\cong\prod_{r=1}^{u}(\mathbb{Z}/p_r^{m_r}\mathbb{Z})^L ( Z / q Z ) L ≅ ∏ r = 1 u ( Z / p r m r Z ) L a ( r ) a^{(r)} a ( r ) a a a p r m r p_r^{m_r} p r m r
( a , q ) = 1 ⟺ ( a ( r ) , p r m r ) = 1 for every 1 ≤ r ≤ u . (a,q)=1\quad\Longleftrightarrow\quad(a^{(r)},p_r^{m_r})=1\quad\text{for every }1\leq r\leq u. ( a , q ) = 1 ⟺ ( a ( r ) , p r m r ) = 1 for every 1 ≤ r ≤ u . Moreover, for every balanced cut S ∣ T S\mid T S ∣ T 2 a S , T 2a_{S,T} 2 a S , T 2 a S , T ( r ) 2a^{(r)}_{S,T} 2 a S , T ( r )
∣ im ( 2 a S , T ) ∣ = ∏ r = 1 u ∣ im ( 2 a S , T ( r ) ) ∣ , \left|\operatorname{im}(2a_{S,T})\right|=\prod_{r=1}^{u}\left|\operatorname{im}(2a^{(r)}_{S,T})\right|, ∣ im ( 2 a S , T ) ∣ = r = 1 ∏ u im ( 2 a S , T ( r ) ) , and consequently s q ( a ) = ∏ r = 1 u s p r m r ( a ( r ) ) . s_q(a)=\prod_{r=1}^{u}s_{p_r^{m_r}}(a^{(r)}). s q ( a ) = ∏ r = 1 u s p r m r ( a ( r ) ) .
∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 s q ( a ) = ∏ r = 1 u ( ∑ a ( r ) ∈ ( Z / p r m r Z ) L ( a ( r ) , p r m r ) = 1 s p r m r ( a ( r ) ) ) ≪ K , d C ω ( q ) ∏ r = 1 u p r − m r ( 1 + 2 η ) = C ω ( q ) q − 1 − 2 η . \sum_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}s_q(a)=\prod_{r=1}^{u}\left(\sum_{\substack{a^{(r)}\in(\mathbb{Z}/p_r^{m_r}\mathbb{Z})^L\\(a^{(r)},p_r^{m_r})=1}}s_{p_r^{m_r}}(a^{(r)})\right)\ll_{K,d}C^{\omega(q)}\prod_{r=1}^{u}p_r^{-m_r(1+2\eta)}=C^{\omega(q)}q^{-1-2\eta}. a ∈ ( Z / q Z ) L ( a , q ) = 1 ∑ s q ( a ) = r = 1 ∏ u a ( r ) ∈ ( Z / p r m r Z ) L ( a ( r ) , p r m r ) = 1 ∑ s p r m r ( a ( r ) ) ≪ K , d C ω ( q ) r = 1 ∏ u p r − m r ( 1 + 2 η ) = C ω ( q ) q − 1 − 2 η . for some C = C ( K , d ) C=C(K,d) C = C ( K , d ) C ω ( q ) ≪ C , η q η C^{\omega(q)}\ll_{C,\eta}q^\eta C ω ( q ) ≪ C , η q η (73) .
Integrating the major arc form
We equip Z d × ( Z / q Z ) d \mathbb{Z}^d\times(\mathbb{Z}/q\mathbb{Z})^d Z d × ( Z / q Z ) d (59) .
Proposition 8.6. Assume d ≥ 4 K + 4 d\geq4K+4 d ≥ 4 K + 4 q ≥ 1 q\geq1 q ≥ 1
G i ∈ L 2 ( Z d × ( Z / q Z ) d ) ( 0 ≤ i ≤ K − 1 ) . G_i\in L^2(\mathbb{Z}^d\times(\mathbb{Z}/q\mathbb{Z})^d)\qquad(0\leq i\leq K-1). G i ∈ L 2 ( Z d × ( Z / q Z ) d ) ( 0 ≤ i ≤ K − 1 ) . For each i i i
supp G i ⊆ [ P ] d × ( Z / q Z ) d . \operatorname{supp}G_i\subseteq[P]^d\times(\mathbb{Z}/q\mathbb{Z})^d. supp G i ⊆ [ P ] d × ( Z / q Z ) d . Then
∑ a m o d q ( a , q ) = 1 ∫ T L ∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ d θ ≪ K , d q − 1 − η P K d / 2 − 2 L ∏ i = 0 K − 1 ∥ G i ∥ 2 . (74) \sum_{\substack{a\bmod q\\(a,q)=1}}\int_{\mathcal{T}_L}\left|\mathcal{M}_{q,a,\theta}(G_0,\ldots,G_{K-1})\right|\,d\theta\ll_{K,d}q^{-1-\eta}P^{Kd/2-2L}\prod_{i=0}^{K-1}\|G_i\|_2. \tag*{(74)} a mod q ( a , q ) = 1 ∑ ∫ T L ∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ d θ ≪ K , d q − 1 − η P K d /2 − 2 L i = 0 ∏ K − 1 ∥ G i ∥ 2 . (74) Proof. Fix a balanced cut S ∣ T S\mid T S ∣ T i i i X i = [ P ] d × ( Z / q Z ) d X_i=[P]^d\times(\mathbb{Z}/q\mathbb{Z})^d X i = [ P ] d × ( Z / q Z ) d
Q S ( u S , r S ) : = Q θ S ( u S ) + Q a S ( r S ) q , Q^S(u_S,r_S):=Q^S_\theta(u_S)+\frac{Q^S_a(r_S)}{q}, Q S ( u S , r S ) := Q θ S ( u S ) + q Q a S ( r S ) , Q S , T ( u S , r S , u T , r T ) : = Q θ S , T ( u S , u T ) + Q a S , T ( r S , r T ) q , Q^{S,T}(u_S,r_S,u_T,r_T):=Q^{S,T}_\theta(u_S,u_T)+\frac{Q^{S,T}_a(r_S,r_T)}{q}, Q S , T ( u S , r S , u T , r T ) := Q θ S , T ( u S , u T ) + q Q a S , T ( r S , r T ) , Q T ( u T , r T ) : = Q θ T ( u T ) + Q a T ( r T ) q . Q^T(u_T,r_T):=Q^T_\theta(u_T)+\frac{Q^T_a(r_T)}{q}. Q T ( u T , r T ) := Q θ T ( u T ) + q Q a T ( r T ) . The cut operator in Lemma 6.1 is therefore
K S , T : L 2 ( ( [ P ] d × ( Z / q Z ) d ) T ) ⟶ L 2 ( ( [ P ] d × ( Z / q Z ) d ) S ) , \mathcal{K}^{S,T}: L^2\left(([P]^d \times(\mathbb{Z}/q\mathbb{Z})^d)^T\right) \longrightarrow L^2\left(([P]^d \times(\mathbb{Z}/q\mathbb{Z})^d)^S\right), K S , T : L 2 ( ([ P ] d × ( Z / q Z ) d ) T ) ⟶ L 2 ( ([ P ] d × ( Z / q Z ) d ) S ) , ( K S , T H ) ( u S , r S ) : = E r T m o d q ∑ u T ∈ ( [ P ] d ) T e q ( Q a S , T ( r S , r T ) ) e ( Q θ S , T ( u S , u T ) ) H ( u T , r T ) . \left(\mathcal{K}^{S,T}H\right)(u_S,r_S) := \mathbb{E}_{r_T \bmod q}\sum_{u_T\in([P]^d)^T}e_q\left(Q_a^{S,T}(r_S,r_T)\right)e\left(Q_\theta^{S,T}(u_S,u_T)\right)H(u_T,r_T). ( K S , T H ) ( u S , r S ) := E r T mod q u T ∈ ([ P ] d ) T ∑ e q ( Q a S , T ( r S , r T ) ) e ( Q θ S , T ( u S , u T ) ) H ( u T , r T ) . Thus
∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ ≤ ∥ K S , T ∥ 2 → 2 ∏ i = 0 K − 1 ∥ G i ∥ 2 . |\mathcal{M}_{q,a,\theta}(G_0,\ldots,G_{K-1})| \le\|\mathcal{K}^{S,T}\|_{2\to2}\prod_{i=0}^{K-1}\|G_i\|_2. ∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ ≤ ∥ K S , T ∥ 2 → 2 i = 0 ∏ K − 1 ∥ G i ∥ 2 . We now identify the norm of K S , T \mathcal{K}^{S,T} K S , T
K S , T = K P , d S , T ( − 2 θ S , T ) ⊗ K q , a , d S , T . (75) \mathcal{K}^{S,T}=\mathcal{K}^{S,T}_{P,d}(-2\theta_{S,T})\otimes\mathcal{K}^{S,T}_{q,a,d}. \tag*{(75)} K S , T = K P , d S , T ( − 2 θ S , T ) ⊗ K q , a , d S , T . (75) Hence (75) , multiplicativity of the operator norm under tensor products, and Lemmas 8.3 and 6.2 give
∥ K S , T ∥ 2 → 2 = ∥ K P , d S , T ( − 2 θ S , T ) ∥ 2 → 2 ∥ K q , a , d S , T ∥ 2 → 2 = P K d / 2 m P S , T ( − 2 θ S , T ) d ∣ im ( 2 a S , T ) ∣ − d / 2 . \begin{aligned}
\|\mathcal{K}^{S,T}\|_{2\to2}=\|\mathcal{K}^{S,T}_{P,d}(-2\theta_{S,T})\|_{2\to2}\|\mathcal{K}^{S,T}_{q,a,d}\|_{2\to2} \\
&=P^{Kd/2}m_P^{S,T}(-2\theta_{S,T})^d|\operatorname{im}(2a_{S,T})|^{-d/2}.
\end{aligned} ∥ K S , T ∥ 2 → 2 = ∥ K P , d S , T ( − 2 θ S , T ) ∥ 2 → 2 ∥ K q , a , d S , T ∥ 2 → 2 = P K d /2 m P S , T ( − 2 θ S , T ) d ∣ im ( 2 a S , T ) ∣ − d /2 . Since d / ∣ B K ∣ = 2 τ w K d/|\mathcal{B}_K|=2\tau w_K d /∣ B K ∣ = 2 τ w K
∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ ≤ P K d / 2 ∏ S ∣ T ∈ B K [ ∣ im ( 2 a S , T ) ∣ − 1 / 2 m P S , T ( − 2 θ S , T ) ] 2 τ w K ∏ i = 0 K − 1 ∥ G i ∥ 2 . |\mathcal{M}_{q,a,\theta}(G_0,\ldots,G_{K-1})|\le P^{Kd/2}\prod_{S|T\in\mathcal{B}_K}\left[|\operatorname{im}(2a_{S,T})|^{-1/2m_P^{S,T}(-2\theta_{S,T})}\right]^{2\tau w_K}\prod_{i=0}^{K-1}\|G_i\|_2. ∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ ≤ P K d /2 S ∣ T ∈ B K ∏ [ ∣ im ( 2 a S , T ) ∣ − 1/2 m P S , T ( − 2 θ S , T ) ] 2 τ w K i = 0 ∏ K − 1 ∥ G i ∥ 2 . By the definition (72) ,
∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ ≤ P K d / 2 s q ( a ) ∏ S ∣ T ∈ B K m P S , T ( − 2 θ S , T ) 2 τ w K ∏ i = 0 K − 1 ∥ G i ∥ 2 . |\mathcal{M}_{q,a,\theta}(G_0,\ldots,G_{K-1})|\le P^{Kd/2}s_q(a)\prod_{S|T\in\mathcal{B}_K}m_P^{S,T}(-2\theta_{S,T})^{2\tau w_K}\prod_{i=0}^{K-1}\|G_i\|_2. ∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ ≤ P K d /2 s q ( a ) S ∣ T ∈ B K ∏ m P S , T ( − 2 θ S , T ) 2 τ w K i = 0 ∏ K − 1 ∥ G i ∥ 2 . It remains to integrate the product of the m P S , T m_P^{S,T} m P S , T J = ∣ B K ∣ J=|\mathcal{B}_K| J = ∣ B K ∣ 1 , … , J 1,\ldots,J 1 , … , J S ∣ T ∈ B K S|T\in\mathcal{B}_K S ∣ T ∈ B K j j j S ∣ T S|T S ∣ T E j \mathcal{E}_j E j S ∣ T S|T S ∣ T c j c_j c j w K w_K w K f S , T f_{S,T} f S , T f j f_j f j
f S , T ( θ S , T ) : = m P S , T ( − 2 θ S , T ) 2 τ . f_{S,T}(\theta_{S,T}):=m_P^{S,T}(-2\theta_{S,T})^{2\tau}. f S , T ( θ S , T ) := m P S , T ( − 2 θ S , T ) 2 τ . For every e ∈ E e\in\mathcal{E} e ∈ E
∑ S ∣ T ∈ B K e c r o s s e s S ∣ T w K = 1 , \sum_{\substack{S|T\in\mathcal{B}_K\\ e\ \mathrm{crosses}\ S|T}}w_K=1, S ∣ T ∈ B K e crosses S ∣ T ∑ w K = 1 , which is exactly the hypothesis of Lemma 6.5. Moreover,
∏ S ∣ T ∈ B K f S , T ( θ S , T ) w K = ∏ S ∣ T ∈ B K m P S , T ( − 2 θ S , T ) 2 τ w K . \prod_{S|T\in\mathcal{B}_K}f_{S,T}(\theta_{S,T})^{w_K}
=\prod_{S|T\in\mathcal{B}_K}m_P^{S,T}(-2\theta_{S,T})^{2\tau w_K}. S ∣ T ∈ B K ∏ f S , T ( θ S , T ) w K = S ∣ T ∈ B K ∏ m P S , T ( − 2 θ S , T ) 2 τ w K . Lemma 6.5 therefore gives
∫ T L ∏ S ∣ T ∈ B K m P S , T ( − 2 θ S , T ) 2 τ w K d θ ≤ ∏ S ∣ T ∈ B K ( ∫ T ∣ S ∣ ∣ T ∣ m P S , T ( − 2 θ S , T ) 2 τ d θ S , T ) w K . \int_{\mathbb{T}_L}\prod_{S|T\in\mathcal{B}_K}m_P^{S,T}(-2\theta_{S,T})^{2\tau w_K}\,d\theta
\le\prod_{S|T\in\mathcal{B}_K}\left(\int_{\mathbb{T}_{|S||T|}}m_P^{S,T}(-2\theta_{S,T})^{2\tau}\,d\theta_{S,T}\right)^{w_K}. ∫ T L S ∣ T ∈ B K ∏ m P S , T ( − 2 θ S , T ) 2 τ w K d θ ≤ S ∣ T ∈ B K ∏ ( ∫ T ∣ S ∣∣ T ∣ m P S , T ( − 2 θ S , T ) 2 τ d θ S , T ) w K . Multiplication by − 2 -2 − 2 T ∣ S ∣ ∣ T ∣ \mathbb{T}^{|S||T|} T ∣ S ∣∣ T ∣ 0 ≤ m P S , T ≤ 1 0 \le m_P^{S,T} \le1 0 ≤ m P S , T ≤ 1 2 τ > 2 K + 4 2\tau> 2K + 4 2 τ > 2 K + 4 Lemma 3.1 gives
∫ T ∣ S ∣ ∣ T ∣ m P S , T ( − 2 θ S , T ) 2 τ d θ S , T ≪ K P − 2 ∣ S ∣ ∣ T ∣ . \int_{\mathbb{T}^{|S||T|}} m_P^{S,T}(-2\theta_{S,T})^{2\tau}\,\mathrm{d}\theta_{S,T} \ll_K P^{-2|S||T|}. ∫ T ∣ S ∣∣ T ∣ m P S , T ( − 2 θ S , T ) 2 τ d θ S , T ≪ K P − 2∣ S ∣∣ T ∣ . Using (55) , we therefore obtain
∫ T L ∏ S ∣ T ∈ B K m P S , T ( − 2 θ S , T ) 2 τ w K d θ ≪ K P − 2 L . \int_{\mathbb{T}^L} \prod_{S|T\in\mathcal{B}_K} m_P^{S,T}(-2\theta_{S,T})^{2\tau w_K}\,\mathrm{d}\theta\ll_K P^{-2L}. ∫ T L S ∣ T ∈ B K ∏ m P S , T ( − 2 θ S , T ) 2 τ w K d θ ≪ K P − 2 L . Summing over a a a Lemma 8.5 and applying this estimate gives
∑ a m o d q ( a , q ) = 1 ∫ T L ∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ d θ ≪ K , d q − 1 − η P K d / 2 − 2 L ∏ i = 0 K − 1 ∥ G i ∥ 2 . \sum_{\substack{a\bmod q\\(a,q)=1}}\int_{\mathbb{T}^L}\left|\mathcal{M}_{q,a,\theta}(G_0,\ldots,G_{K-1})\right|\,\mathrm{d}\theta\ll_{K,d} q^{-1-\eta}P^{Kd/2-2L}\prod_{i=0}^{K-1}\|G_i\|_2. a mod q ( a , q ) = 1 ∑ ∫ T L ∣ M q , a , θ ( G 0 , … , G K − 1 ) ∣ d θ ≪ K , d q − 1 − η P K d /2 − 2 L i = 0 ∏ K − 1 ∥ G i ∥ 2 . This is (74) .
Proof of the major arc estimate
We can now prove the major arc estimate stated at the beginning of the section. It will simplify the argument to cover M Q \mathfrak{M}_Q M Q a / q a/q a / q q q q 1 ≤ q ≤ 2 Q L 1\le q\le2Q^L 1 ≤ q ≤ 2 Q L a ∈ ( Z / q Z ) L a\in(\mathbb{Z}/q\mathbb{Z})^L a ∈ ( Z / q Z ) L ( a , q ) = 1 (a,q)=1 ( a , q ) = 1
B Q ( a / q ) : = ∏ e ∈ E { β e ∈ T : ∥ β e − a e / q ∥ T ≤ Q R 2 } . \mathfrak{B}_Q(a/q):=\prod_{e\in\mathcal{E}}\left\{\beta_e\in\mathbb{T}:\|\beta_e-a_e/q\|_{\mathbb{T}}\le\frac{Q}{R^2}\right\}. B Q ( a / q ) := e ∈ E ∏ { β e ∈ T : ∥ β e − a e / q ∥ T ≤ R 2 Q } . Lemma 8.7. Assume 2 ≤ Q ≤ R 2\le Q\le R 2 ≤ Q ≤ R Then
M Q ⊆ ⋃ 1 ≤ q ≤ 2 Q L ⋃ a ∈ ( Z / q Z ) L ( a , q ) = 1 B Q ( a / q ) . (76) \mathfrak{M}_Q\subseteq\bigcup_{1\le q\le2Q^L}\ \bigcup_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}\mathfrak{B}_Q(a/q). \tag*{(76)} M Q ⊆ 1 ≤ q ≤ 2 Q L ⋃ a ∈ ( Z / q Z ) L ( a , q ) = 1 ⋃ B Q ( a / q ) . (76) Proof. Fix β ∈ M Q \beta\in\mathfrak{M}_Q β ∈ M Q e ∈ E e\in\mathcal{E} e ∈ E 1 ≤ q e ≤ Q 1\le q_e\le Q 1 ≤ q e ≤ Q b e ∈ Z / q e Z b_e\in\mathbb{Z}/q_e\mathbb{Z} b e ∈ Z / q e Z ( b e , q e ) = 1 (b_e,q_e)=1 ( b e , q e ) = 1
∥ − 2 β e − b e / q e ∥ T ≤ Q q e R 2 . \|-2\beta_e-b_e/q_e\|_{\mathbb{T}}\le\frac{Q}{q_eR^2}. ∥ − 2 β e − b e / q e ∥ T ≤ q e R 2 Q . The two preimages of b e / q e b_e/q_e b e / q e x ↦ − 2 x x\mapsto-2x x ↦ − 2 x − b e / ( 2 q e ) -b_e/(2q_e) − b e / ( 2 q e ) ( q e − b e ) / ( 2 q e ) (q_e-b_e)/(2q_e) ( q e − b e ) / ( 2 q e ) β e \beta_e β e ρ e \rho_e ρ e − 2 ρ e = b e / q e ∈ T -2\rho_e=b_e/q_e\in\mathbb{T} − 2 ρ e = b e / q e ∈ T
∥ β e − ρ e ∥ T = 1 2 ∥ − 2 β e − b e / q e ∥ T ≤ Q 2 q e R 2 . \|\beta_e-\rho_e\|_{\mathbb{T}}=\frac{1}{2}\|-2\beta_e-b_e/q_e\|_{\mathbb{T}}\le\frac{Q}{2q_eR^2}. ∥ β e − ρ e ∥ T = 2 1 ∥ − 2 β e − b e / q e ∥ T ≤ 2 q e R 2 Q . Notice that the ρ e \rho_e ρ e 2 q e 2q_e 2 q e
Put ρ : = ( ρ e ) e ∈ E \rho:=(\rho_e)_{e\in\mathcal{E}} ρ := ( ρ e ) e ∈ E q q q
q ∣ lcm e ∈ E ( 2 q e ) = 2 lcm e ∈ E q e , q\mid\operatorname{lcm}_{e\in\mathcal{E}}(2q_e)=2\operatorname{lcm}_{e\in\mathcal{E}}q_e, q ∣ lcm e ∈ E ( 2 q e ) = 2 lcm e ∈ E q e , and hence
q ≤ 2 ∏ e ∈ E q e ≤ 2 Q L . q\le2\prod_{e\in\mathcal{E}}q_e\le2Q^L. q ≤ 2 e ∈ E ∏ q e ≤ 2 Q L . Write ρ = a / q \rho=a/q ρ = a / q a ∈ ( Z / q Z ) L a\in(\mathbb{Z}/q\mathbb{Z})^{L} a ∈ ( Z / q Z ) L q q q ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 e ∈ E e\in\mathcal{E} e ∈ E
∥ β e − a e / q ∥ T = ∥ β e − ρ e ∥ T ≤ Q 2 q e R 2 ≤ Q R 2 . \lVert\beta_e-a_e/q\rVert_{\mathbb{T}}=\lVert\beta_e-\rho_e\rVert_{\mathbb{T}}\le\frac{Q}{2q_eR^2}\le\frac{Q}{R^2}. ∥ β e − a e / q ∥ T = ∥ β e − ρ e ∥ T ≤ 2 q e R 2 Q ≤ R 2 Q . Therefore β ∈ B Q ( a / q ) \beta\in\mathcal{B}_Q(a/q) β ∈ B Q ( a / q ) (76) .
Proof of Proposition 8.1. By definition,
M N Q ( F 0 , … , F K − 1 ) = ∫ M Q w ^ R ( Q β ( v ) ) S β ( F 0 , … , F K − 1 ) d β . \mathfrak{M}_N^Q(F_0,\ldots,F_{K-1})=\int_{\mathfrak{M}_Q}\widehat{w}_R(\mathcal{Q}_\beta(v))\mathcal{S}_\beta(F_0,\ldots,F_{K-1})\,\mathrm{d}\beta. M N Q ( F 0 , … , F K − 1 ) = ∫ M Q w R ( Q β ( v )) S β ( F 0 , … , F K − 1 ) d β . Lemma 8.7, the triangle inequality, and (32) therefore give
∣ M N Q ( F 0 , … , F K − 1 ) ∣ ≪ Δ ∑ 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∫ [ − Q / R 2 , Q / R 2 ] L ∣ S a / q + θ ( F 0 , … , F K − 1 ) ∣ d θ . (77) \left|\mathfrak{M}_N^Q(F_0,\ldots,F_{K-1})\right|\ll_{\Delta}\sum_{1\le q\le2Q^L}\sum_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}\int_{[-Q/R^2,Q/R^2]^L}\left|\mathcal{S}_{a/q+\theta}(F_0,\ldots,F_{K-1})\right|\,\mathrm{d}\theta. \tag*{(77)} M N Q ( F 0 , … , F K − 1 ) ≪ Δ 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∑ ∫ [ − Q / R 2 , Q / R 2 ] L S a / q + θ ( F 0 , … , F K − 1 ) d θ . (77) For every q q q a a a θ \theta θ (77) , put
E q , a , θ : = S a / q + θ ( F 0 , … , F K − 1 ) − M q , a , θ ( G 0 , q , … , G K − 1 , q ) . E_{q,a,\theta}:=\mathcal{S}_{a/q+\theta}(F_0,\ldots,F_{K-1})-\mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q}). E q , a , θ := S a / q + θ ( F 0 , … , F K − 1 ) − M q , a , θ ( G 0 , q , … , G K − 1 , q ) . Lemma 8.2 gives
∣ E q , a , θ ∣ ≪ K , d , σ Q − L ( L + 2 ) − 2 N K d / 2 ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (78) \left|E_{q,a,\theta}\right|\ll_{K,d,\sigma}Q^{-L(L+2)-2}N^{Kd/2}\prod_{i=0}^{K-1}\lVert F_i\rVert_{\ell^2(\mathbb{Z}^d)}. \tag*{(78)} ∣ E q , a , θ ∣ ≪ K , d , σ Q − L ( L + 2 ) − 2 N K d /2 i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (78) Put
P q : = N + q N q + q . (79) P_q:=N+qN_q+q. \tag*{(79)} P q := N + q N q + q . (79) Then
supp G i , q ⊆ [ P q ] d × ( Z / q Z ) d . \operatorname{supp}G_{i,q}\subseteq[P_q]^d\times(\mathbb{Z}/q\mathbb{Z})^d. supp G i , q ⊆ [ P q ] d × ( Z / q Z ) d . Indeed, if G i , q ( u , r ) ≠ 0 G_{i,q}(u,r)\ne0 G i , q ( u , r ) = 0 (60) and the definition of μ q \mu_q μ q z ∈ [ N ] d z\in[N]^d z ∈ [ N ] d h ∈ { 0 , … , N q − 1 } d h\in\{0,\ldots,N_q-1\}^d h ∈ { 0 , … , N q − 1 } d s ∈ { 0 , … , q − 1 } d s\in\{0,\ldots,q-1\}^d s ∈ { 0 , … , q − 1 } d
F i ( z ) ≠ 0 , u = z + q h + s . F_i(z)\ne0,\qquad u=z+qh+s. F i ( z ) = 0 , u = z + q h + s . Proposition 8.6 therefore gives
∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∫ [ − Q / R 2 , Q / R 2 ] L ∣ M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ d θ ≤ ∑ a m o d q ( a , q ) = 1 ∫ T L ∣ M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ d θ \sum_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}\int_{[-Q/R^2,Q/R^2]^L}\left|\mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})\right|\,\mathrm{d}\theta\le\sum_{\substack{a\bmod q\\(a,q)=1}}\int_{\mathbb{T}^L}\left|\mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})\right|\,\mathrm{d}\theta a ∈ ( Z / q Z ) L ( a , q ) = 1 ∑ ∫ [ − Q / R 2 , Q / R 2 ] L ∣ M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ d θ ≤ a mod q ( a , q ) = 1 ∑ ∫ T L ∣ M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ d θ ≪ K , d q − 1 − η P q K d / 2 − 2 L ∏ i = 0 K − 1 ∥ G i , q ∥ 2 . (80) \ll_{K,d}q^{-1-\eta}P_q^{Kd/2-2L}\prod_{i=0}^{K-1}\lVert G_{i,q}\rVert_2. \tag*{(80)} ≪ K , d q − 1 − η P q K d /2 − 2 L i = 0 ∏ K − 1 ∥ G i , q ∥ 2 . (80) Observe that
∥ G i , q ∥ 2 ≤ P q d ( 1 / 2 − 1 / K ) ∥ G i , q ∥ K . (81) \lVert G_{i,q}\rVert_2\le P_q^{d(1/2-1/K)}\lVert G_{i,q}\rVert_K. \tag*{(81)} ∥ G i , q ∥ 2 ≤ P q d ( 1/2 − 1/ K ) ∥ G i , q ∥ K . (81) For fixed u ∈ Z d u\in\mathbb{Z}^d u ∈ Z d y = y ( u , r ) y=y(u,r) y = y ( u , r ) (60) . The map r ⟼ s : = u − y ( u , r ) r\longmapsto s:=u-y(u,r) r ⟼ s := u − y ( u , r ) ( Z / q Z ) d (\mathbb{Z}/q\mathbb{Z})^d ( Z / q Z ) d { 0 , … , q − 1 } d \{0,\ldots,q-1\}^d { 0 , … , q − 1 } d
∥ G i , q ∥ K K = ∑ u ∈ Z d E r m o d q ∣ G i , q ( u , r ) ∣ K = 1 q d ∑ s ∈ { 0 , … , q − 1 } d ∑ u ∈ Z d ∣ ( μ q ∗ F i ) ( u − s ) ∣ K = ∥ μ q ∗ F i ∥ ℓ K ( Z d ) K . (82) \begin{aligned}
\lVert G_{i,q}\rVert_K^K=\sum_{u\in\mathbb{Z}^d}\mathbb{E}_{r\mathop{\rm mod}q}\left|G_{i,q}(u,r)\right|^K \\
&=\frac{1}{q^d}\sum_{s\in\{0,\ldots,q-1\}^d}\sum_{u\in\mathbb{Z}^d}\left|(\mu_q*F_i)(u-s)\right|^K \\
&=\lVert\mu_q*F_i\rVert_{\ell^K(\mathbb{Z}^d)}^K. \tag*{(82)}
\end{aligned} ∥ G i , q ∥ K K = u ∈ Z d ∑ E r mod q ∣ G i , q ( u , r ) ∣ K = q d 1 s ∈ { 0 , … , q − 1 } d ∑ u ∈ Z d ∑ ∣ ( μ q ∗ F i ) ( u − s ) ∣ K = ∥ μ q ∗ F i ∥ ℓ K ( Z d ) K . (82) Moreover,
q N q ≤ R , q ≤ 2 Q L ≤ R , N ≤ 2 σ − 1 R . qN_q \le R,\qquad q \le2Q^L \le R,\qquad N \le2\sigma^{-1}R. q N q ≤ R , q ≤ 2 Q L ≤ R , N ≤ 2 σ − 1 R . Here the first inequality follows from (57) , the second from (56) , and the third from R = ⌊ σ N ⌋ R=\lfloor\sigma N\rfloor R = ⌊ σ N ⌋ N ≥ 2 / σ N\ge2/\sigma N ≥ 2/ σ P q ≪ σ R P_q\ll_\sigma R P q ≪ σ R (81) and (82) give
P q K d / 2 − 2 L ∏ i = 0 K − 1 ∥ G i , q ∥ 2 ≤ P q ( K − 1 ) d − 2 L ∏ i = 0 K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) . P_q^{Kd/2-2L}\prod_{i=0}^{K-1}\|G_{i,q}\|_2
\le P_q^{(K-1)d-2L}\prod_{i=0}^{K-1}\|\mu_q*F_i\|_{\ell^K(\mathbb{Z}^d)}. P q K d /2 − 2 L i = 0 ∏ K − 1 ∥ G i , q ∥ 2 ≤ P q ( K − 1 ) d − 2 L i = 0 ∏ K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) . Consequently, summing (80) over q ≤ 2 Q L q\le2Q^L q ≤ 2 Q L
∑ 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∫ [ − Q / R 2 , Q / R 2 ] L ∣ M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ d θ ≪ K , d , σ R ( K − 1 ) d − 2 L ∑ q ≤ 2 Q L q − 1 − η ∏ i = 0 K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) . (83) \sum_{1\le q\le2Q^L}\ \sum_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}\int_{[-Q/R^2,Q/R^2]^L}\left|\mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})\right|\,d\theta
\ll_{K,d,\sigma}R^{(K-1)d-2L}\sum_{q\le2Q^L}q^{-1-\eta}\prod_{i=0}^{K-1}\|\mu_q*F_i\|_{\ell^K(\mathbb{Z}^d)}. \tag*{(83)} 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∑ ∫ [ − Q / R 2 , Q / R 2 ] L ∣ M q , a , θ ( G 0 , q , … , G K − 1 , q ) ∣ d θ ≪ K , d , σ R ( K − 1 ) d − 2 L q ≤ 2 Q L ∑ q − 1 − η i = 0 ∏ K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) . (83) For each q q q
# { a ∈ ( Z / q Z ) L : ( a , q ) = 1 } ≤ q L , \#\left\{a\in(\mathbb{Z}/q\mathbb{Z})^L:(a,q)=1\right\}\le q^L, # { a ∈ ( Z / q Z ) L : ( a , q ) = 1 } ≤ q L , and the measure of [ − Q / R 2 , Q / R 2 ] L [-Q/R^2,Q/R^2]^L [ − Q / R 2 , Q / R 2 ] L ( 2 Q / R 2 ) L (2Q/R^2)^L ( 2 Q / R 2 ) L
∑ 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∫ [ − Q / R 2 , Q / R 2 ] L d θ ≤ ( 2 Q R 2 ) L ∑ q ≤ 2 Q L q L ≪ L Q L ( L + 2 ) R − 2 L . \sum_{1\le q\le2Q^L}\ \sum_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}\int_{[-Q/R^2,Q/R^2]^L}d\theta
\le\left(\frac{2Q}{R^2}\right)^L\sum_{q\le2Q^L}q^L
\ll_L Q^{L(L+2)}R^{-2L}. 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∑ ∫ [ − Q / R 2 , Q / R 2 ] L d θ ≤ ( R 2 2 Q ) L q ≤ 2 Q L ∑ q L ≪ L Q L ( L + 2 ) R − 2 L . It follows from (78) that
∑ 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∫ [ − Q / R 2 , Q / R 2 ] L ∣ E q , a , θ ∣ d θ ≪ K , d , σ Q − L ( L + 2 ) − 2 N K d / 2 ⋅ Q L ( L + 2 ) R − 2 L ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) \sum_{1\le q\le2Q^L}\ \sum_{\substack{a\in(\mathbb{Z}/q\mathbb{Z})^L\\(a,q)=1}}\int_{[-Q/R^2,Q/R^2]^L}|E_{q,a,\theta}|\,d\theta
\ll_{K,d,\sigma}Q^{-L(L+2)-2}N^{Kd/2}\cdot Q^{L(L+2)}R^{-2L}\prod_{i=0}^{K-1}\|F_i\|_{\ell^2(\mathbb{Z}^d)} 1 ≤ q ≤ 2 Q L ∑ a ∈ ( Z / q Z ) L ( a , q ) = 1 ∑ ∫ [ − Q / R 2 , Q / R 2 ] L ∣ E q , a , θ ∣ d θ ≪ K , d , σ Q − L ( L + 2 ) − 2 N K d /2 ⋅ Q L ( L + 2 ) R − 2 L i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) ≪ K , d , σ Q − 2 R K d / 2 − 2 L ∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (84) \ll_{K,d,\sigma}Q^{-2}R^{Kd/2-2L}\prod_{i=0}^{K-1}\|F_i\|_{\ell^2(\mathbb{Z}^d)}. \tag*{(84)} ≪ K , d , σ Q − 2 R K d /2 − 2 L i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) . (84) Since
S a / q + θ ( F 0 , … , F K − 1 ) = M q , a , θ ( G 0 , q , … , G K − 1 , q ) + E q , a , θ , S_{a/q+\theta}(F_0,\ldots,F_{K-1})=\mathcal{M}_{q,a,\theta}(G_{0,q},\ldots,G_{K-1,q})+E_{q,a,\theta}, S a / q + θ ( F 0 , … , F K − 1 ) = M q , a , θ ( G 0 , q , … , G K − 1 , q ) + E q , a , θ , the bounds (77) , (83) , and (84) prove (58) .
The first term in (58) retains the averages μ q ∗ F i \mu_q*F_i μ q ∗ F i 0 < γ ≤ 1 0<\gamma\le1 0 < γ ≤ 1 Q − 2 Q^{-2} Q − 2 O ( Q − γ ) O(Q^{-\gamma}) O ( Q − γ )
The density increment and completion of the proof
The following proposition gives the dichotomy that will be iterated in the proof of Theorem 1.1.
Proposition 9.1 (Density-increment). For the constant σ \sigma σ 0 < κ < 1 0<\kappa<1 0 < κ < 1 B 0 ≥ 1 B_0\ge1 B 0 ≥ 1 Δ \Delta Δ d d d A ⊆ [ N ] d A\subseteq[N]^d A ⊆ [ N ] d Δ \Delta Δ α : = ∣ A ∣ / N d \alpha:=|A|/N^d α := ∣ A ∣/ N d
(1) N ≪ Δ , d α − B 0 N \ll_{\Delta,d} \alpha^{-B_0} N ≪ Δ , d α − B 0
(2) There are an integer N ′ ≥ 1 N' \ge1 N ′ ≥ 1 and a set A ′ ⊆ [ N ′ ] d A' \subseteq[N']^d A ′ ⊆ [ N ′ ] d , also containing no nondegenerate similar copy of Δ \Delta Δ , such that N ′ ≫ Δ , d α B 0 N N' \gg_{\Delta,d} \alpha^{B_0}N N ′ ≫ Δ , d α B 0 N and ∣ A ′ ∣ / ( N ′ ) d > ( 1 + κ ) α |A'|/(N')^d > (1+\kappa)\alpha ∣ A ′ ∣/ ( N ′ ) d > ( 1 + κ ) α .
Proof. Let B 0 ≥ 1 B_0 \ge1 B 0 ≥ 1 Q 0 ≥ 2 Q_0 \ge2 Q 0 ≥ 2 0 < κ < 1 0 < \kappa< 1 0 < κ < 1 Δ \Delta Δ d d d Q : = Q 0 α − B 0 / ( L ( L + 3 ) + 3 ) Q := Q_0\alpha^{-B_0/(L(L+3)+3)} Q := Q 0 α − B 0 / ( L ( L + 3 ) + 3 ) N < N 0 N < N_0 N < N 0 R ≥ σ N / 2 R \ge\sigma N/2 R ≥ σ N /2 R < 4 Q L ( L + 3 ) + 3 R < 4Q^{L(L+3)+3} R < 4 Q L ( L + 3 ) + 3
N ≪ Δ , d Q L ( L + 3 ) + 3 = Q 0 L ( L + 3 ) + 3 α − B 0 ≪ Δ , d α − B 0 , N \ll_{\Delta,d} Q^{L(L+3)+3}=Q_0^{L(L+3)+3}\alpha^{-B_0}\ll_{\Delta,d}\alpha^{-B_0}, N ≪ Δ , d Q L ( L + 3 ) + 3 = Q 0 L ( L + 3 ) + 3 α − B 0 ≪ Δ , d α − B 0 , so alternative (1) again holds. We may therefore assume (56) . In particular, 2 ≤ Q ≤ R 2 \le Q \le R 2 ≤ Q ≤ R
Let B \mathcal{B} B s = ∣ B ∣ s=|\mathcal{B}| s = ∣ B ∣ (28) , and take F i = g F_i=g F i = g i ∈ B i\in\mathcal{B} i ∈ B F i = 1 [ N ] d F_i=\mathbf{1}_{[N]^d} F i = 1 [ N ] d
α s N d R ( K − 1 ) d − 2 L ≪ Δ , d ∣ Λ N M Q ( F 0 , … , F K − 1 ) ∣ + ∣ Λ N m Q ( F 0 , … , F K − 1 ) ∣ . \alpha^sN^dR^{(K-1)d-2L}\ll_{\Delta,d}|\Lambda_N^{\mathfrak{M}_Q}(F_0,\ldots,F_{K-1})|+|\Lambda_N^{\mathfrak{m}_Q}(F_0,\ldots,F_{K-1})|. α s N d R ( K − 1 ) d − 2 L ≪ Δ , d ∣ Λ N M Q ( F 0 , … , F K − 1 ) ∣ + ∣ Λ N m Q ( F 0 , … , F K − 1 ) ∣. The bounds in (26) give
∏ i = 0 K − 1 ∥ F i ∥ ℓ 2 ( Z d ) = ∥ g ∥ ℓ 2 ( Z d ) s ∥ 1 [ N ] d ∥ ℓ 2 ( Z d ) K − s ≤ α s / 2 N K d / 2 . \prod_{i=0}^{K-1}\|F_i\|_{\ell^2(\mathbb{Z}^d)}
=\|g\|_{\ell^2(\mathbb{Z}^d)}^s\|\mathbf{1}_{[N]^d}\|_{\ell^2(\mathbb{Z}^d)}^{K-s}
\le\alpha^{s/2}N^{Kd/2}. i = 0 ∏ K − 1 ∥ F i ∥ ℓ 2 ( Z d ) = ∥ g ∥ ℓ 2 ( Z d ) s ∥ 1 [ N ] d ∥ ℓ 2 ( Z d ) K − s ≤ α s /2 N K d /2 . Since μ q \mu_q μ q
∥ μ q ∗ 1 [ N ] d ∥ ℓ K ( Z d ) ≤ ∥ μ q ∥ ℓ 1 ( Z d ) ∥ 1 [ N ] d ∥ ℓ K ( Z d ) = N d / K . \|\mu_q * \mathbf{1}_{[N]^d}\|_{\ell^K(\mathbb{Z}^d)}
\le\|\mu_q\|_{\ell^1(\mathbb{Z}^d)}\|\mathbf{1}_{[N]^d}\|_{\ell^K(\mathbb{Z}^d)}
=N^{d/K}. ∥ μ q ∗ 1 [ N ] d ∥ ℓ K ( Z d ) ≤ ∥ μ q ∥ ℓ 1 ( Z d ) ∥ 1 [ N ] d ∥ ℓ K ( Z d ) = N d / K . Consequently,
∏ i = 0 K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) = ∥ μ q ∗ g ∥ ℓ K ( Z d ) s ∥ μ q ∗ 1 [ N ] d ∥ ℓ K ( Z d ) K − s ≤ N d ( K − s ) / K ( ∑ x ∈ Z d ∣ μ q ∗ g ( x ) ∣ K ) s / K = N d ( 1 N d ∑ x ∈ Z d ∣ μ q ∗ g ( x ) ∣ K ) s / K , \begin{aligned}
\prod_{i=0}^{K-1}\|\mu_q * F_i\|_{\ell^K(\mathbb{Z}^d)}
&=\|\mu_q*g\|_{\ell^K(\mathbb{Z}^d)}^s
\|\mu_q*\mathbf{1}_{[N]^d}\|_{\ell^K(\mathbb{Z}^d)}^{K-s}\\
&\le N^{d(K-s)/K}
\left(\sum_{x\in\mathbb{Z}^d}|\mu_q*g(x)|^K\right)^{s/K}\\
&=N^d\left(\frac{1}{N^d}\sum_{x\in\mathbb{Z}^d}|\mu_q*g(x)|^K\right)^{s/K},
\end{aligned} i = 0 ∏ K − 1 ∥ μ q ∗ F i ∥ ℓ K ( Z d ) = ∥ μ q ∗ g ∥ ℓ K ( Z d ) s ∥ μ q ∗ 1 [ N ] d ∥ ℓ K ( Z d ) K − s ≤ N d ( K − s ) / K ( x ∈ Z d ∑ ∣ μ q ∗ g ( x ) ∣ K ) s / K = N d ( N d 1 x ∈ Z d ∑ ∣ μ q ∗ g ( x ) ∣ K ) s / K , where d ( K − s ) / K + d s / K = d d(K-s)/K+ds/K=d d ( K − s ) / K + d s / K = d
Since R ≍ Δ N R\asymp_{\Delta}N R ≍ Δ N Q − 2 ≤ Q − γ Q^{-2}\le Q^{-\gamma} Q − 2 ≤ Q − γ
α s ≪ Δ , d ∑ q ≤ 2 Q L q − 1 − η ( 1 N d ∑ x ∈ Z d ∣ μ q ∗ g ( x ) ∣ K ) s / K + Q − γ α s / 2 . (85) \alpha^s\ll_{\Delta,d}\sum_{q\le2Q^L}q^{-1-\eta}
\left(\frac{1}{N^d}\sum_{x\in\mathbb{Z}^d}|\mu_q*g(x)|^K\right)^{s/K}+Q^{-\gamma}\alpha^{s/2}. \tag*{(85)} α s ≪ Δ , d q ≤ 2 Q L ∑ q − 1 − η ( N d 1 x ∈ Z d ∑ ∣ μ q ∗ g ( x ) ∣ K ) s / K + Q − γ α s /2 . (85) Assuming B 0 B_0 B 0 K K K d d d α \alpha α s s s
Q − γ α s / 2 = Q 0 − γ α γ B 0 / ( L ( L + 3 ) + 3 ) + s / 2 ≤ Q 0 − γ α s . Q^{-\gamma}\alpha^{s/2}
=Q_0^{-\gamma}\alpha^{\gamma B_0/(L(L+3)+3)+s/2}
\le Q_0^{-\gamma}\alpha^s. Q − γ α s /2 = Q 0 − γ α γ B 0 / ( L ( L + 3 ) + 3 ) + s /2 ≤ Q 0 − γ α s . The contribution of this term to (85) is therefore O Δ , d ( Q 0 − γ α s ) O_{\Delta,d}(Q_0^{-\gamma}\alpha^s) O Δ , d ( Q 0 − γ α s ) Q 0 Q_0 Q 0 Δ \Delta Δ d d d α s / 2 \alpha^s/2 α s /2 ∑ q ≥ 1 q − 1 − η < ∞ \sum_{q\ge1}q^{-1-\eta}<\infty ∑ q ≥ 1 q − 1 − η < ∞ q ≤ 2 Q L q\le2Q^L q ≤ 2 Q L
∑ x ∈ Z d ∣ μ q ∗ g ( x ) ∣ K ≫ Δ , d α K N d . (86) \sum_{x\in\mathbb{Z}^d}|\mu_q*g(x)|^K\gg_{\Delta,d}\alpha^KN^d. \tag*{(86)} x ∈ Z d ∑ ∣ μ q ∗ g ( x ) ∣ K ≫ Δ , d α K N d . (86) Recall that N q = ⌊ R / ( q Q 0 L ( L + 2 ) + 3 ) ⌋ N_q=\left\lfloor R/(qQ_0^{L(L+2)+3})\right\rfloor N q = ⌊ R / ( q Q 0 L ( L + 2 ) + 3 ) ⌋ μ q ∗ g \mu_q * g μ q ∗ g [ 1 , N + q ( N q − 1 ) ] d [1,N+q(N_q-1)]^d [ 1 , N + q ( N q − 1 ) ] d
∣ [ 1 , N + q ( N q − 1 ) ] d ∖ [ 1 + q ( N q − 1 ) , N ] d ∣ ≪ d q N q N d − 1 ≪ Δ , d Q 0 − L ( L + 2 ) − 3 N d ≪ Δ , d Q 0 − L ( L + 2 ) − 3 α B 0 ( L ( L + 2 ) + 3 ) / ( L ( L + 3 ) + 3 ) N d . \left|[1,N+q(N_q-1)]^d\setminus[1+q(N_q-1),N]^d\right| \ll_d qN_qN^{d-1}
\ll_{\Delta,d} Q_0^{-L(L+2)-3}N^d
\ll_{\Delta,d} Q_0^{-L(L+2)-3} \alpha^{B_0(L(L+2)+3)/(L(L+3)+3)}N^d. [ 1 , N + q ( N q − 1 ) ] d ∖ [ 1 + q ( N q − 1 ) , N ] d ≪ d q N q N d − 1 ≪ Δ , d Q 0 − L ( L + 2 ) − 3 N d ≪ Δ , d Q 0 − L ( L + 2 ) − 3 α B 0 ( L ( L + 2 ) + 3 ) / ( L ( L + 3 ) + 3 ) N d . Assuming B 0 B_0 B 0 K K K α \alpha α K K K
∣ [ 1 , N + q ( N q − 1 ) ] d ∖ [ 1 + q ( N q − 1 ) , N ] d ∣ ≪ Δ , d Q 0 − L ( L + 2 ) − 3 α K N d . (87) \left|[1,N+q(N_q-1)]^d\setminus[1+q(N_q-1),N]^d\right| \ll_{\Delta,d} Q_0^{-L(L+2)-3}\alpha^K N^d. \tag*{(87)} [ 1 , N + q ( N q − 1 ) ] d ∖ [ 1 + q ( N q − 1 ) , N ] d ≪ Δ , d Q 0 − L ( L + 2 ) − 3 α K N d . (87) Since μ q \mu_q μ q ∑ x ∈ Z d g ( x ) = 0 \sum_{x\in\mathbb{Z}^d}g(x)=0 ∑ x ∈ Z d g ( x ) = 0 ∑ x ∈ Z d ( μ q ∗ g ) ( x ) = 0 \sum_{x\in\mathbb{Z}^d}(\mu_q*g)(x)=0 ∑ x ∈ Z d ( μ q ∗ g ) ( x ) = 0
∑ x ∈ Z d ( μ q ∗ g ) + ( x ) = ∑ x ∈ Z d ( μ q ∗ g ) − ( x ) = 1 2 ∑ x ∈ Z d ∣ ( μ q ∗ g ) ( x ) ∣ . (88) \sum_{x\in\mathbb{Z}^d}(\mu_q*g)_+(x)=\sum_{x\in\mathbb{Z}^d}(\mu_q*g)_-(x)=\frac{1}{2}\sum_{x\in\mathbb{Z}^d}|(\mu_q*g)(x)|. \tag*{(88)} x ∈ Z d ∑ ( μ q ∗ g ) + ( x ) = x ∈ Z d ∑ ( μ q ∗ g ) − ( x ) = 2 1 x ∈ Z d ∑ ∣ ( μ q ∗ g ) ( x ) ∣. (88) Since g ≥ − α g\ge-\alpha g ≥ − α ∣ g ∣ ≤ 1 |g|\le1 ∣ g ∣ ≤ 1
0 ≤ ( μ q ∗ g ) − ≤ α , ∣ μ q ∗ g ∣ ≤ 1. 0\le(\mu_q*g)_-\le\alpha,\qquad|\mu_q*g|\le1. 0 ≤ ( μ q ∗ g ) − ≤ α , ∣ μ q ∗ g ∣ ≤ 1. Assuming Q 0 Q_0 Q 0 Δ \Delta Δ d d d (87) and (86) give
∑ x ∈ [ 1 + q ( N q − 1 ) , N ] d ∣ ( μ q ∗ g ) ( x ) ∣ K ≫ Δ , d α K N d . (89) \sum_{x\in[1+q(N_q-1),N]^d}|(\mu_q*g)(x)|^K\gg_{\Delta,d}\alpha^K N^d. \tag*{(89)} x ∈ [ 1 + q ( N q − 1 ) , N ] d ∑ ∣ ( μ q ∗ g ) ( x ) ∣ K ≫ Δ , d α K N d . (89) On [ 1 + q ( N q − 1 ) , N ] d [1+q(N_q-1),N]^d [ 1 + q ( N q − 1 ) , N ] d μ q ∗ g = μ q ∗ 1 A − α \mu_q*g=\mu_q*1_A-\alpha μ q ∗ g = μ q ∗ 1 A − α
∣ ( μ q ∗ g ) ( x ) ∣ K = ( ( μ q ∗ g ) + ( x ) ) K + ( ( μ q ∗ g ) − ( x ) ) K , |(\mu_q*g)(x)|^K=((\mu_q*g)_+(x))^K+((\mu_q*g)_-(x))^K, ∣ ( μ q ∗ g ) ( x ) ∣ K = (( μ q ∗ g ) + ( x ) ) K + (( μ q ∗ g ) − ( x ) ) K , (89) implies that at least one of the following two estimates holds:
∑ x ∈ [ 1 + q ( N q − 1 ) , N ] d ( ( μ q ∗ g ) + ( x ) ) K ≫ Δ , d α K N d , (90) \sum_{x\in[1+q(N_q-1),N]^d}((\mu_q*g)_+(x))^K\gg_{\Delta,d}\alpha^K N^d, \tag*{(90)} x ∈ [ 1 + q ( N q − 1 ) , N ] d ∑ (( μ q ∗ g ) + ( x ) ) K ≫ Δ , d α K N d , (90) ∑ x ∈ [ 1 + q ( N q − 1 ) , N ] d ( ( μ q ∗ g ) − ( x ) ) K ≫ Δ , d α K N d . (91) \sum_{x\in[1+q(N_q-1),N]^d}((\mu_q*g)_-(x))^K\gg_{\Delta,d}\alpha^K N^d. \tag*{(91)} x ∈ [ 1 + q ( N q − 1 ) , N ] d ∑ (( μ q ∗ g ) − ( x ) ) K ≫ Δ , d α K N d . (91) If (90) holds, pigeonholing over at most N d N^d N d y ∈ [ 1 + q ( N q − 1 ) , N ] d y\in[1+q(N_q-1),N]^d y ∈ [ 1 + q ( N q − 1 ) , N ] d ( μ q ∗ g ) ( y ) ≫ Δ , d α (\mu_q*g)(y)\gg_{\Delta,d}\alpha ( μ q ∗ g ) ( y ) ≫ Δ , d α (91) holds. Since ( μ q ∗ g ) − ≤ α (\mu_q*g)_-\le\alpha ( μ q ∗ g ) − ≤ α
∑ x ∈ [ 1 + q ( N q − 1 ) , N ] d ( μ q ∗ g ) − ( x ) ≥ α 1 − K ∑ x ∈ [ 1 + q ( N q − 1 ) , N ] d ( ( μ q ∗ g ) − ( x ) ) K ≫ Δ , d α N d . \sum_{x\in[1+q(N_q-1),N]^d}(\mu_q*g)_-(x)\ge\alpha^{1-K}\sum_{x\in[1+q(N_q-1),N]^d}((\mu_q*g)_-(x))^K\gg_{\Delta,d}\alpha N^d. x ∈ [ 1 + q ( N q − 1 ) , N ] d ∑ ( μ q ∗ g ) − ( x ) ≥ α 1 − K x ∈ [ 1 + q ( N q − 1 ) , N ] d ∑ (( μ q ∗ g ) − ( x ) ) K ≫ Δ , d α N d . It follows from (88) that
∑ x ∈ Z d ( μ q ∗ g ) + ( x ) = ∑ x ∈ Z d ( μ q ∗ g ) − ( x ) ≫ Δ , d α N d . (92) \sum_{x\in\mathbb{Z}^d}(\mu_q*g)_+(x)=\sum_{x\in\mathbb{Z}^d}(\mu_q*g)_-(x)\gg_{\Delta,d}\alpha N^d. \tag*{(92)} x ∈ Z d ∑ ( μ q ∗ g ) + ( x ) = x ∈ Z d ∑ ( μ q ∗ g ) − ( x ) ≫ Δ , d α N d . (92) Since μ q ∗ g \mu_q*g μ q ∗ g [ 1 , N + q ( N q − 1 ) ] d [1,N+q(N_q-1)]^d [ 1 , N + q ( N q − 1 ) ] d ( μ q ∗ g ) + ≤ 1 (\mu_q*g)_+\le1 ( μ q ∗ g ) + ≤ 1 (87) gives
∑ x ∈ [ 1 , N + q ( N q − 1 ) ] d ∖ [ 1 + q ( N q − 1 ) , N ] d ( μ q ∗ g ) + ( x ) = O Δ , d ( Q 0 − L ( L + 2 ) − 3 α K N d ) . \sum_{x\in[1,N+q(N_q-1)]^d\setminus[1+q(N_q-1),N]^d}(\mu_q*g)_+(x)
=O_{\Delta,d}\left(Q_0^{-L(L+2)-3}\alpha^K N^d\right). x ∈ [ 1 , N + q ( N q − 1 ) ] d ∖ [ 1 + q ( N q − 1 ) , N ] d ∑ ( μ q ∗ g ) + ( x ) = O Δ , d ( Q 0 − L ( L + 2 ) − 3 α K N d ) . Assuming Q 0 Q_0 Q 0 Δ \Delta Δ d d d (92) , and α K ≤ α \alpha^K \le\alpha α K ≤ α
∑ x ∈ [ 1 + q ( N q − 1 ) , N ] d ( μ q ∗ g ) ( x ) ≫ Δ , d α N d . \sum_{x\in[1+q(N_q-1),N]^d}(\mu_q * g)(x) \gg_{\Delta,d} \alpha N^d. x ∈ [ 1 + q ( N q − 1 ) , N ] d ∑ ( μ q ∗ g ) ( x ) ≫ Δ , d α N d . Pigeonholing over at most N d N^d N d y ∈ [ 1 + q ( N q − 1 ) , N ] d y\in[1+q(N_q-1),N]^d y ∈ [ 1 + q ( N q − 1 ) , N ] d ( μ q ∗ g ) ( y ) ≫ Δ , d α (\mu_q * g)(y)\gg_{\Delta,d}\alpha ( μ q ∗ g ) ( y ) ≫ Δ , d α κ \kappa κ Δ \Delta Δ d d d y ∈ [ 1 + q ( N q − 1 ) , N ] d y\in[1+q(N_q-1),N]^d y ∈ [ 1 + q ( N q − 1 ) , N ] d
( μ q ∗ 1 A ) ( y ) = ( μ q ∗ g ) ( y ) + α > ( 1 + κ ) α . (\mu_q * 1_A)(y)=(\mu_q * g)(y)+\alpha>(1+\kappa)\alpha. ( μ q ∗ 1 A ) ( y ) = ( μ q ∗ g ) ( y ) + α > ( 1 + κ ) α . Set N ′ : = N q N':=N_q N ′ := N q q ≤ 2 Q L q\le2Q^L q ≤ 2 Q L (56) holds,
R q Q L ( L + 2 ) + 3 ≥ R 2 Q L ( L + 3 ) + 3 ≥ 2. \frac{R}{qQ^{L(L+2)+3}}\ge\frac{R}{2Q^{L(L+3)+3}}\ge2. q Q L ( L + 2 ) + 3 R ≥ 2 Q L ( L + 3 ) + 3 R ≥ 2. It follows that
N ′ = ⌊ R q Q L ( L + 2 ) + 3 ⌋ ≥ R 2 q Q L ( L + 2 ) + 3 ≥ R 4 Q L ( L + 3 ) + 3 ≫ Δ , d α B 0 N . N'=\left\lfloor\frac{R}{qQ^{L(L+2)+3}}\right\rfloor\ge\frac{R}{2qQ^{L(L+2)+3}}\ge\frac{R}{4Q^{L(L+3)+3}}\gg_{\Delta,d}\alpha^{B_0}N. N ′ = ⌊ q Q L ( L + 2 ) + 3 R ⌋ ≥ 2 q Q L ( L + 2 ) + 3 R ≥ 4 Q L ( L + 3 ) + 3 R ≫ Δ , d α B 0 N . Finally, put
A ′ : = { n ∈ [ N ′ ] d : y − q ( n − ( 1 , … , 1 ) ) ∈ A } . A':=\left\{n\in[N']^d:y-q(n-(1,\ldots,1))\in A\right\}. A ′ := { n ∈ [ N ′ ] d : y − q ( n − ( 1 , … , 1 )) ∈ A } . Because y ∈ [ 1 + q ( N q − 1 ) , N ] d y\in[1+q(N_q-1),N]^d y ∈ [ 1 + q ( N q − 1 ) , N ] d N ′ = N q N'=N_q N ′ = N q n ↦ y − q ( n − ( 1 , … , 1 ) ) n\mapsto y-q(n-(1,\ldots,1)) n ↦ y − q ( n − ( 1 , … , 1 )) [ N ′ ] d [N']^d [ N ′ ] d [ N ] d [N]^d [ N ] d μ q \mu_q μ q
∣ A ′ ∣ ( N ′ ) d = ( μ q ∗ 1 A ) ( y ) > ( 1 + κ ) α . \frac{|A'|}{(N')^d}=(\mu_q * 1_A)(y)>(1+\kappa)\alpha. ( N ′ ) d ∣ A ′ ∣ = ( μ q ∗ 1 A ) ( y ) > ( 1 + κ ) α . The map n ↦ y − q ( n − ( 1 , … , 1 ) ) n\mapsto y-q(n-(1,\ldots,1)) n ↦ y − q ( n − ( 1 , … , 1 )) A ′ A' A ′ Δ \Delta Δ
Proof of Theorem 1.1. Fix the constants supplied by Proposition 9.1. If A = ∅ A=\varnothing A = ∅ A 0 = A A_0=A A 0 = A N 0 = N N_0=N N 0 = N j j j α j : = ∣ A j ∣ / N j d \alpha_j:=|A_j|/N_j^d α j := ∣ A j ∣/ N j d A j + 1 ⊆ [ N j + 1 ] d A_{j+1}\subseteq[N_{j+1}]^d A j + 1 ⊆ [ N j + 1 ] d 1 + κ 1+\kappa 1 + κ 1 1 1 j ∗ j_* j ∗
N j ∗ ≪ Δ , d α j ∗ − B 0 . (93) N_{j_*}\ll_{\Delta,d}\alpha_{j_*}^{-B_0}. \tag*{(93)} N j ∗ ≪ Δ , d α j ∗ − B 0 . (93) Iterating the density increase in alternative (2) gives 1 ≥ α j ∗ ≥ ( 1 + κ ) j ∗ α 0 1\ge\alpha_{j_*}\ge(1+\kappa)^{j_*}\alpha_0 1 ≥ α j ∗ ≥ ( 1 + κ ) j ∗ α 0 j ∗ ≪ Δ , d 1 + log ( 1 / α 0 ) j_*\ll_{\Delta,d}1+\log(1/\alpha_0) j ∗ ≪ Δ , d 1 + log ( 1/ α 0 ) N j + 1 ≫ Δ , d α j B 0 N j N_{j+1}\gg_{\Delta,d}\alpha_j^{B_0}N_j N j + 1 ≫ Δ , d α j B 0 N j 0 ≤ j < j ∗ 0\le j<j_* 0 ≤ j < j ∗
log N j ≤ log N j + 1 + B 0 log ( 1 / α j ) + O Δ , d ( 1 ) . \log N_j\le\log N_{j+1}+B_0\log(1/\alpha_j)+O_{\Delta,d}(1). log N j ≤ log N j + 1 + B 0 log ( 1/ α j ) + O Δ , d ( 1 ) . Summing over j j j (93) , we obtain
log N 0 ≤ B 0 ∑ j = 0 j ∗ log ( 1 / α j ) + O Δ , d ( j ∗ + 1 ) . \log N_0\le B_0\sum_{j=0}^{j_*}\log(1/\alpha_j)+O_{\Delta,d}(j_*+1). log N 0 ≤ B 0 j = 0 ∑ j ∗ log ( 1/ α j ) + O Δ , d ( j ∗ + 1 ) . Since α j ≥ α 0 \alpha_j\ge\alpha_0 α j ≥ α 0 j j j
log N 0 ≪ Δ , d ( 1 + log ( 1 / α 0 ) ) 2 . \log N_0\ll_{\Delta,d}\left(1+\log(1/\alpha_0)\right)^2. log N 0 ≪ Δ , d ( 1 + log ( 1/ α 0 ) ) 2 . Consequently, for suitable constants C , c > 0 C,c>0 C , c > 0 Δ \Delta Δ d d d
α 0 ≤ C exp ( − c log N 0 ) . \alpha_0\le C\exp\left(-c\sqrt{\log N_0}\right). α 0 ≤ C exp ( − c log N 0 ) . Since α 0 = ∣ A ∣ / N 0 d \alpha_0=|A|/N_0^d α 0 = ∣ A ∣/ N 0 d
Application to a continuous variant of the problem
Proof of Corollary 1.3. Let C 0 , c 0 > 0 C_0,c_0>0 C 0 , c 0 > 0 Theorem 1.1 , and put L α : = 1 + log ( 1 / α ) L_\alpha:= 1+\log(1/\alpha) L α := 1 + log ( 1/ α ) D = D Δ , d D=D_{\Delta,d} D = D Δ , d
N : = ⌈ exp ( D L α 2 ) ⌉ . N:=\left\lceil\exp(DL_\alpha^2)\right\rceil. N := ⌈ exp ( D L α 2 ) ⌉ . Then C 0 exp ( − c 0 log N ) < α C_0\exp(-c_0\sqrt{\log N})<\alpha C 0 exp ( − c 0 log N ) < α N ≤ 2 exp ( D L α 2 ) N\le2\exp(DL_\alpha^2) N ≤ 2 exp ( D L α 2 ) u ∈ [ 0 , 1 / N ) d u\in[0,1/N)^d u ∈ [ 0 , 1/ N ) d
B u : = { m ∈ { 0 , … , N − 1 } d : u + m N ∈ E } . B_u:=\left\{m\in\{0,\ldots,N-1\}^d:u+\frac{m}{N}\in E\right\}. B u := { m ∈ { 0 , … , N − 1 } d : u + N m ∈ E } . The translates of [ 0 , 1 / N ) d [0,1/N)^d [ 0 , 1/ N ) d m / N m/N m / N [ 0 , 1 ) d [0,1)^d [ 0 , 1 ) d
∫ [ 0 , 1 / N ) d ∣ B u ∣ d u = ∣ E ∣ . \int_{[0,1/N)^d}|B_u|\,du=|E|. ∫ [ 0 , 1/ N ) d ∣ B u ∣ d u = ∣ E ∣. There is consequently some u u u ∣ B u ∣ ≥ α N d |B_u|\ge\alpha N^d ∣ B u ∣ ≥ α N d N N N Theorem 1.1 , the translate B u + ( 1 , … , 1 ) ⊆ [ N ] d B_u+(1,\ldots,1)\subseteq[N]^d B u + ( 1 , … , 1 ) ⊆ [ N ] d Δ \Delta Δ y 0 , … , y K − 1 ∈ B u y_0,\ldots,y_{K-1}\in B_u y 0 , … , y K − 1 ∈ B u ρ > 0 \rho>0 ρ > 0
For any fixed edge { i , j } \{i,j\} { i , j } Δ \Delta Δ y i − y j y_i-y_j y i − y j
ρ 2 = ∣ y i − y j ∣ 2 ∣ v i − v j ∣ 2 ≥ 1 ∣ v i − v j ∣ 2 , \rho^2=\frac{|y_i-y_j|^2}{|v_i-v_j|^2}\ge\frac{1}{|v_i-v_j|^2}, ρ 2 = ∣ v i − v j ∣ 2 ∣ y i − y j ∣ 2 ≥ ∣ v i − v j ∣ 2 1 , so ρ ≥ c Δ > 0 \rho\ge c_\Delta>0 ρ ≥ c Δ > 0 E E E m ↦ u + m / N m\mapsto u+m/N m ↦ u + m / N
λ = ρ N ≥ exp ( − C Δ , d ( 1 + log ( 1 / α ) ) 2 ) , \lambda=\frac{\rho}{N}\ge\exp\left(-C_{\Delta,d}(1+\log(1/\alpha))^2\right), λ = N ρ ≥ exp ( − C Δ , d ( 1 + log ( 1/ α ) ) 2 ) , after enlarging C Δ , d C_{\Delta,d} C Δ , d x + λ U Δ x+\lambda U\Delta x + λ U Δ x ∈ R d x\in\mathbb{R}^d x ∈ R d U ∈ O ( d ) U\in O(d) U ∈ O ( d )
Counting solutions to the matrix equation
The trace expansion in Lemma 3.1 reduces the required high-moment estimate to counting bounded integer matrices U U U V V V U V T = 0 UV^{\mathsf T}=0 U V T = 0
Lemma B.1 (Integer matrix orthogonality). Let s , t , n s,t,n s , t , n
n ≥ 2 max ( s , t ) and n > s + t . n\ge2\max(s,t)\qquad\text{and}\qquad n>s+t. n ≥ 2 max ( s , t ) and n > s + t . Then, for every positive integer R R R
# { ( U , V ) : U ∈ ( [ − R , R ] ∩ Z ) s × n , V ∈ ( [ − R , R ] ∩ Z ) t × n , U V T = 0 } ≪ n , s , t R n ( s + t ) − 2 s t . \#\left\{(U,V):
\begin{array}{l}
U\in([-R,R]\cap\mathbb{Z})^{s\times n},\\
V\in([-R,R]\cap\mathbb{Z})^{t\times n},\\
UV^{\mathsf T}=0
\end{array}
\right\}\ll_{n,s,t}R^{n(s+t)-2st}. # ⎩ ⎨ ⎧ ( U , V ) : U ∈ ([ − R , R ] ∩ Z ) s × n , V ∈ ([ − R , R ] ∩ Z ) t × n , U V T = 0 ⎭ ⎬ ⎫ ≪ n , s , t R n ( s + t ) − 2 s t . In the application, n = K + 1 n=K+1 n = K + 1 s = ∣ S ∣ s=|S| s = ∣ S ∣ t = ∣ T ∣ t=|T| t = ∣ T ∣ n > s + t n>s+t n > s + t ∣ S ∣ + ∣ T ∣ = K |S|+|T|=K ∣ S ∣ + ∣ T ∣ = K n ≥ 2 max ( s , t ) n\ge2\max(s,t) n ≥ 2 max ( s , t ) S ∣ T S\mid T S ∣ T
Lattice preliminaries
The definitions and elementary determinant facts below are standard; see [2 Chapter I] . We follow [16 §1] for the determinant convention and for the notation X n X_n X n μ n \mu_n μ n Gr ( L , d ) \operatorname{Gr}(L,d) Gr ( L , d )
A lattice in R n \mathbb{R}^n R n Γ = Z γ 1 + ⋯ + Z γ d \Gamma= \mathbb{Z}\gamma_1+\cdots+\mathbb{Z}\gamma_d Γ = Z γ 1 + ⋯ + Z γ d γ 1 , … , γ d ∈ R n \gamma_1,\ldots,\gamma_d \in\mathbb{R}^n γ 1 , … , γ d ∈ R n Γ \Gamma Γ d d d Γ \Gamma Γ
span R Γ : = R γ 1 + ⋯ + R γ d . \operatorname{span}_{\mathbb{R}}\Gamma:= \mathbb{R}\gamma_1+\cdots+\mathbb{R}\gamma_d. span R Γ := R γ 1 + ⋯ + R γ d . A sublattice of a lattice L L L Γ ⊆ L \Gamma\subseteq L Γ ⊆ L γ 1 , … , γ d \gamma_1,\ldots,\gamma_d γ 1 , … , γ d
G ( γ 1 , … , γ d ) : = ( ⟨ γ i , γ j ⟩ ) i , j = 1 d . G(\gamma_1,\ldots,\gamma_d):=\left(\langle\gamma_i,\gamma_j\rangle\right)_{i,j=1}^{d}. G ( γ 1 , … , γ d ) := ( ⟨ γ i , γ j ⟩ ) i , j = 1 d . Following Kim, define the determinant of Γ \Gamma Γ
det Γ : = ∥ γ 1 ∧ ⋯ ∧ γ d ∥ = ( det G ( γ 1 , … , γ d ) ) 1 / 2 . \det\Gamma:=\lVert\gamma_1\wedge\cdots\wedge\gamma_d\rVert=\left(\det G(\gamma_1,\ldots,\gamma_d)\right)^{1/2}. det Γ := ∥ γ 1 ∧ ⋯ ∧ γ d ∥ = ( det G ( γ 1 , … , γ d ) ) 1/2 . Any two bases of Γ \Gamma Γ A ∈ GL d ( Z ) A\in\operatorname{GL}_d(\mathbb{Z}) A ∈ GL d ( Z ) G G G A G A T AGA^{\mathsf{T}} A G A T det G \det G det G det Γ \det\Gamma det Γ
If Γ ⊆ Λ \Gamma\subseteq\Lambda Γ ⊆ Λ [ Λ : Γ ] [\Lambda:\Gamma] [ Λ : Γ ] Γ \Gamma Γ Λ \Lambda Λ
det Γ = [ Λ : Γ ] det Λ . (94) \det\Gamma=[\Lambda:\Gamma]\det\Lambda. \tag*{(94)} det Γ = [ Λ : Γ ] det Λ. (94) Thus enlarging a lattice without changing its rank can only decrease its determinant. Moreover, if Γ ⊆ Z n \Gamma\subseteq\mathbb{Z}^n Γ ⊆ Z n Γ \Gamma Γ
det Γ ≥ 1. (95) \det\Gamma\geq1. \tag*{(95)} det Γ ≥ 1. (95) A lattice in R n \mathbb{R}^n R n n n n g ∈ GL n ( R ) g\in\operatorname{GL}_n(\mathbb{R}) g ∈ GL n ( R ) Z n g : = { z g : z ∈ Z n } \mathbb{Z}^n g:=\{zg:z\in\mathbb{Z}^n\} Z n g := { z g : z ∈ Z n } ∣ det g ∣ |\det g| ∣ det g ∣ Z n g \mathbb{Z}^n g Z n g g ∈ SL ( n , R ) g\in\operatorname{SL}(n,\mathbb{R}) g ∈ SL ( n , R )
Following Kim, let X n : = SL ( n , Z ) \ SL ( n , R ) X_n:=\operatorname{SL}(n,\mathbb{Z})\backslash\operatorname{SL}(n,\mathbb{R}) X n := SL ( n , Z ) \ SL ( n , R ) g ∈ SL ( n , R ) g\in\operatorname{SL}(n,\mathbb{R}) g ∈ SL ( n , R ) Z n g \mathbb{Z}^n g Z n g SL ( n , Z ) \operatorname{SL}(n,\mathbb{Z}) SL ( n , Z ) X n X_n X n R n \mathbb{R}^n R n μ n \mu_n μ n SL ( n , R ) \operatorname{SL}(n,\mathbb{R}) SL ( n , R )
A sublattice Γ ⊆ L \Gamma\subseteq L Γ ⊆ L L L L Γ = L ∩ span R Γ \Gamma=L\cap\operatorname{span}_{\mathbb{R}}\Gamma Γ = L ∩ span R Γ Γ ⊆ L \Gamma\subseteq L Γ ⊆ L L ∩ span R Γ L\cap\operatorname{span}_{\mathbb{R}}\Gamma L ∩ span R Γ Γ \Gamma Γ L L L L L L Γ \Gamma Γ
For L ∈ X n L\in X_n L ∈ X n 1 ≤ d < n 1\leq d<n 1 ≤ d < n Gr ( L , d ) \operatorname{Gr}(L,d) Gr ( L , d ) d d d L L L
For a sublattice Γ ⊆ Z n \Gamma\subseteq\mathbb{Z}^n Γ ⊆ Z n
Γ ⊥ Z : = { x ∈ Z n : ⟨ x , γ ⟩ = 0 for every γ ∈ Γ } = ( span R Γ ) ⊥ ∩ Z n . \Gamma^{\perp_{\mathbb{Z}}:=\{x\in\mathbb{Z}^n:\langle x,\gamma\rangle=0\text{ for every }\gamma\in\Gamma\}=(\operatorname{span}_{\mathbb{R}}\Gamma)^\perp\cap\mathbb{Z}^n.} Γ ⊥ Z := { x ∈ Z n : ⟨ x , γ ⟩ = 0 for every γ ∈ Γ } = ( span R Γ ) ⊥ ∩ Z n . If Γ ⊆ Z n \Gamma\subseteq\mathbb{Z}^n Γ ⊆ Z n d d d [27 §2, especially Lemma 1 and its corollary] gives
Γ ⊥ Z is primitive of rank n − d , (96) \Gamma^{\perp_{\mathbb{Z}}}\text{ is primitive of rank }n-d, \tag*{(96)} Γ ⊥ Z is primitive of rank n − d , (96) det ( Γ ⊥ Z ) = det Γ , (97) \det(\Gamma^{\perp\mathbb{Z}})=\det\Gamma, \tag*{(97)} det ( Γ ⊥ Z ) = det Γ , (97) and
( Γ ⊥ Z ) ⊥ Z = Γ . (98) (\Gamma^{\perp\mathbb{Z}})^{\perp\mathbb{Z}}=\Gamma. \tag*{(98)} ( Γ ⊥ Z ) ⊥ Z = Γ. (98) For a rank-d d d Γ \Gamma Γ 1 ≤ j ≤ d 1 \le j \le d 1 ≤ j ≤ d j j j λ j ( Γ ) \lambda_j(\Gamma) λ j ( Γ ) λ > 0 \lambda> 0 λ > 0 Γ \Gamma Γ j j j λ \lambda λ λ 1 ( Γ ) ≤ ⋯ ≤ λ d ( Γ ) \lambda_1(\Gamma) \le\cdots\le\lambda_d(\Gamma) λ 1 ( Γ ) ≤ ⋯ ≤ λ d ( Γ ) [27 §5, Lemma 2] is
# ( Γ ∩ [ − R , R ] n ) ≪ n ∏ j = 1 d ( 1 + R λ j ( Γ ) ) , (99) \#(\Gamma\cap[-R,R]^n) \ll_n \prod_{j=1}^{d}\left(1+\frac{R}{\lambda_j(\Gamma)}\right), \tag*{(99)} # ( Γ ∩ [ − R , R ] n ) ≪ n j = 1 ∏ d ( 1 + λ j ( Γ ) R ) , (99) and Minkowski’s second theorem [2 Chapter VIII, Theorem V] gives
∏ j = 1 d λ j ( Γ ) ≍ n det Γ . (100) \prod_{j=1}^{d}\lambda_j(\Gamma)\asymp_n\det\Gamma. \tag*{(100)} j = 1 ∏ d λ j ( Γ ) ≍ n det Γ. (100) If λ d ( Γ ) ≪ n R \lambda_d(\Gamma)\ll_n R λ d ( Γ ) ≪ n R 1 + R / λ j ( Γ ) ≪ n R / λ j ( Γ ) 1+R/\lambda_j(\Gamma)\ll_n R/\lambda_j(\Gamma) 1 + R / λ j ( Γ ) ≪ n R / λ j ( Γ ) 1 ≤ j ≤ d 1\le j\le d 1 ≤ j ≤ d
# ( Γ ∩ [ − R , R ] n ) ≪ n R d det Γ . (101) \#(\Gamma\cap[-R,R]^n)\ll_n\frac{R^d}{\det\Gamma}. \tag*{(101)} # ( Γ ∩ [ − R , R ] n ) ≪ n det Γ R d . (101) Finally, we fix Kim’s notation for the flags appearing below. Let V ( i ) : = π i / 2 / Γ ( 1 + i / 2 ) V(i):=\pi^{i/2}/\Gamma(1+i/2) V ( i ) := π i /2 /Γ ( 1 + i /2 ) R i \mathbb{R}^i R i ζ ( 1 ) = 1 \zeta(1)=1 ζ ( 1 ) = 1 1 ≤ d < n 1\le d<n 1 ≤ d < n
a ( n , d ) : = 1 n ( n d ) ∏ i = 1 d V ( n − i + 1 ) ζ ( i ) V ( i ) ζ ( n − i + 1 ) . a(n,d):=\frac{1}{n}\binom{n}{d}\prod_{i=1}^{d}\frac{V(n-i+1)\zeta(i)}{V(i)\zeta(n-i+1)}. a ( n , d ) := n 1 ( d n ) i = 1 ∏ d V ( i ) ζ ( n − i + 1 ) V ( n − i + 1 ) ζ ( i ) . For
d 0 = 0 < d 1 < ⋯ < d k < d k + 1 = n , d = ( d 1 , … , d k ) , d_0=0<d_1<\cdots<d_k<d_{k+1}=n,\qquad\boldsymbol{d}=(d_1,\ldots,d_k), d 0 = 0 < d 1 < ⋯ < d k < d k + 1 = n , d = ( d 1 , … , d k ) , Kim calls a chain
A 1 ⊆ ⋯ ⊆ A k ⊆ L , A i ∈ Gr ( L , d i ) , A_1\subseteq\cdots\subseteq A_k\subseteq L,\qquad A_i\in\operatorname{Gr}(L,d_i), A 1 ⊆ ⋯ ⊆ A k ⊆ L , A i ∈ Gr ( L , d i ) , a flag of type d \boldsymbol{d} d L L L
a ( n , d ) : = a ( n , d 1 ) ∏ i = 1 k − 1 n − d i − 1 d i + 1 − d i − 1 a ( n − d i , d i + 1 − d i ) . a(n,\boldsymbol{d}):=a(n,d_1)\prod_{i=1}^{k-1}\frac{n-d_{i-1}}{d_{i+1}-d_{i-1}}a(n-d_i,d_{i+1}-d_i). a ( n , d ) := a ( n , d 1 ) i = 1 ∏ k − 1 d i + 1 − d i − 1 n − d i − 1 a ( n − d i , d i + 1 − d i ) . Kim’s mean value formula for primitive lattice flags
We now record the only non-elementary input.
Theorem B.2 (Kim’s mean value formula for rational flags [16 Theorem 5] ). For H 1 , … , H k ≥ 0 H_1,\ldots,H_k\ge0 H 1 , … , H k ≥ 0 μ n \mu_n μ n A 1 ⊂ ⋯ ⊂ A k A_1\subset\cdots\subset A_k A 1 ⊂ ⋯ ⊂ A k d \boldsymbol{d} d L L L det A i ≤ H i \det A_i\le H_i det A i ≤ H i i = 1 , … , k i=1,\ldots,k i = 1 , … , k a ( n , d ) ∏ i = 1 k H i d i + 1 − d i − 1 a(n,\boldsymbol{d})\prod_{i=1}^{k}H_i^{d_{i+1}-d_{i-1}} a ( n , d ) ∏ i = 1 k H i d i + 1 − d i − 1
We transfer the case k = 2 k=2 k = 2 Z n \mathbb{Z}^n Z n
Lemma B.3 (Two-height flag count). Fix positive integers n , d 1 , d 2 n,d_1,d_2 n , d 1 , d 2 0 < d 1 < d 2 < n 0<d_1<d_2<n 0 < d 1 < d 2 < n H 1 , H 2 ≥ 1 H_1,H_2\ge1 H 1 , H 2 ≥ 1
# { ( A 1 , A 2 ) : A 1 ∈ Gr ( Z n , d 1 ) , A 2 ∈ Gr ( Z n , d 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 } ≪ n , d 1 , d 2 H 1 d 2 H 2 n − d 1 . \#\left\{(A_1,A_2):
\begin{array}{l}
A_1\in\operatorname{Gr}(\mathbb{Z}^n,d_1),\\
A_2\in\operatorname{Gr}(\mathbb{Z}^n,d_2),\\
A_1\subseteq A_2,\\
\det A_1\le H_1,\quad\det A_2\le H_2
\end{array}
\right\}
\ll_{n,d_1,d_2}H_1^{d_2}H_2^{n-d_1}. # ⎩ ⎨ ⎧ ( A 1 , A 2 ) : A 1 ∈ Gr ( Z n , d 1 ) , A 2 ∈ Gr ( Z n , d 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 ⎭ ⎬ ⎫ ≪ n , d 1 , d 2 H 1 d 2 H 2 n − d 1 . Proof. Take k = 2 k=2 k = 2 d = ( d 1 , d 2 ) \mathbf{d}=(d_1,d_2) d = ( d 1 , d 2 ) Theorem B.2 . Since d 0 = 0 d_0=0 d 0 = 0 d 3 = n d_3=n d 3 = n
∫ X n # { ( A 1 , A 2 ) : A 1 ∈ Gr ( L , d 1 ) , A 2 ∈ Gr ( L , d 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 } d μ n ( L ) = a ( n , d ) H 1 d 2 − d 0 H 2 d 3 − d 1 = a ( n , d ) H 1 d 2 H 2 n − d 1 . (102) \begin{aligned}
\int_{X_n}\#\left\{(A_1,A_2):
\begin{aligned}
&A_1\in\operatorname{Gr}(L,d_1),\\
&A_2\in\operatorname{Gr}(L,d_2),\\
&A_1\subseteq A_2,\\
&\det A_1\le H_1,\quad\det A_2\le H_2
\end{aligned}
\right\}\,d\mu_n(L)
=a(n,\mathbf{d})H_1^{d_2-d_0}H_2^{d_3-d_1} \\
&=a(n,\mathbf{d})H_1^{d_2}H_2^{n-d_1}. \tag*{(102)}
\end{aligned} ∫ X n # ⎩ ⎨ ⎧ ( A 1 , A 2 ) : A 1 ∈ Gr ( L , d 1 ) , A 2 ∈ Gr ( L , d 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 ⎭ ⎬ ⎫ d μ n ( L ) = a ( n , d ) H 1 d 2 − d 0 H 2 d 3 − d 1 = a ( n , d ) H 1 d 2 H 2 n − d 1 . (102) It remains to pass from this average to the particular lattice Z n \mathbb{Z}^n Z n π : SL ( n , R ) → X n \pi:\operatorname{SL}(n,\mathbb{R})\to X_n π : SL ( n , R ) → X n W W W π \pi π U = π ( W ) U=\pi(W) U = π ( W ) μ n ( U ) > 0 \mu_n(U)>0 μ n ( U ) > 0 A ⊆ Z n A\subseteq\mathbb{Z}^n A ⊆ Z n g ∈ SL ( n , R ) g\in\operatorname{SL}(n,\mathbb{R}) g ∈ SL ( n , R ) A g : = { a g : a ∈ A } A_g:=\{ag:a\in A\} A g := { a g : a ∈ A } 1 ≤ j < n 1\le j<n 1 ≤ j < n W W W c j < ∞ c_j<\infty c j < ∞
det ( A q ) ≤ c j det A ( A ∈ Gr ( Z n , j ) , q ∈ W ) . \det(Aq)\le c_j\det A\qquad(A\in\operatorname{Gr}(\mathbb{Z}^n,j),\ q\in W). det ( A q ) ≤ c j det A ( A ∈ Gr ( Z n , j ) , q ∈ W ) . Right multiplication by g g g Gr ( Z n , j ) \operatorname{Gr}(\mathbb{Z}^n,j) Gr ( Z n , j ) Gr ( Z n g , j ) \operatorname{Gr}(\mathbb{Z}^n g,j) Gr ( Z n g , j ) g ∈ W g\in W g ∈ W
# { ( A 1 , A 2 ) : A i ∈ Gr ( Z n , d i ) ( i = 1 , 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 } ≤ # { ( A 1 , A 2 ) : A i ∈ Gr ( Z n g , d i ) ( i = 1 , 2 ) , A 1 ⊆ A 2 , det A 1 ≤ c d 1 H 1 , det A 2 ≤ c d 2 H 2 } . \#\left\{(A_1,A_2):
\begin{aligned}
&A_i\in\operatorname{Gr}(\mathbb{Z}^n,d_i)\quad(i=1,2),\\
&A_1\subseteq A_2,\\
&\det A_1\le H_1,\quad\det A_2\le H_2
\end{aligned}
\right\}
\le
\#\left\{(A_1,A_2):
\begin{aligned}
&A_i\in\operatorname{Gr}(\mathbb{Z}^n g,d_i)\quad(i=1,2),\\
&A_1\subseteq A_2,\\
&\det A_1\le c_{d_1}H_1,\\
&\det A_2\le c_{d_2}H_2
\end{aligned}
\right\}. # ⎩ ⎨ ⎧ ( A 1 , A 2 ) : A i ∈ Gr ( Z n , d i ) ( i = 1 , 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 ⎭ ⎬ ⎫ ≤ # ⎩ ⎨ ⎧ ( A 1 , A 2 ) : A i ∈ Gr ( Z n g , d i ) ( i = 1 , 2 ) , A 1 ⊆ A 2 , det A 1 ≤ c d 1 H 1 , det A 2 ≤ c d 2 H 2 ⎭ ⎬ ⎫ . Integrating this inequality over U U U ( H 1 , H 2 ) (H_1,H_2) ( H 1 , H 2 ) ( c d 1 H 1 , c d 2 H 2 ) (c_{d_1}H_1,c_{d_2}H_2) ( c d 1 H 1 , c d 2 H 2 )
μ n ( U ) # { ( A 1 , A 2 ) : A i ∈ Gr ( Z n , d i ) ( i = 1 , 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 } ≤ a ( n , d ) ( c d 1 H 1 ) d 2 ( c d 2 H 2 ) n − d 1 . \mu_n(U)\#\left\{(A_1,A_2):
\begin{aligned}
&A_i\in\operatorname{Gr}(\mathbb{Z}^n,d_i)\quad(i=1,2),\\
&A_1\subseteq A_2,\\
&\det A_1\le H_1,\quad\det A_2\le H_2
\end{aligned}
\right\}
\le a(n,\mathbf{d})(c_{d_1}H_1)^{d_2}(c_{d_2}H_2)^{n-d_1}. μ n ( U ) # ⎩ ⎨ ⎧ ( A 1 , A 2 ) : A i ∈ Gr ( Z n , d i ) ( i = 1 , 2 ) , A 1 ⊆ A 2 , det A 1 ≤ H 1 , det A 2 ≤ H 2 ⎭ ⎬ ⎫ ≤ a ( n , d ) ( c d 1 H 1 ) d 2 ( c d 2 H 2 ) n − d 1 . Since W W W n n n □ \square □
Proof of the matrix count
Proof of Lemma B.1. Decompose the count according to
d 1 = rank U , d 2 = n − rank V . d_1=\operatorname{rank} U,\qquad d_2=n-\operatorname{rank} V. d 1 = rank U , d 2 = n − rank V . If d 1 = 0 d_1=0 d 1 = 0 U = 0 U=0 U = 0 O ( R n t ) O(R^{nt}) O ( R n t ) V V V
n ( s + t ) − 2 s t − n t = s ( n − 2 t ) ≥ 0. n(s+t)-2st-nt=s(n-2t)\ge0. n ( s + t ) − 2 s t − n t = s ( n − 2 t ) ≥ 0. The case d 2 = n d_2=n d 2 = n V = 0 V=0 V = 0
1 ≤ d 1 ≤ s , n − t ≤ d 2 ≤ n − 1. 1\le d_1\le s,\qquad n-t\le d_2\le n-1. 1 ≤ d 1 ≤ s , n − t ≤ d 2 ≤ n − 1. Since d 1 + ( n − d 2 ) ≤ s + t < n d_1+(n-d_2)\le s+t<n d 1 + ( n − d 2 ) ≤ s + t < n d 1 < d 2 d_1<d_2 d 1 < d 2
The rows of U U U Z n \mathbb{Z}^n Z n Z n \mathbb{Z}^n Z n V V V
Γ U : = rowspan R ( U ) ∩ Z n , Γ V : = rowspan R ( V ) ∩ Z n . \Gamma_U:=\operatorname{rowspan}_{\mathbb{R}}(U)\cap\mathbb{Z}^n,\qquad
\Gamma_V:=\operatorname{rowspan}_{\mathbb{R}}(V)\cap\mathbb{Z}^n. Γ U := rowspan R ( U ) ∩ Z n , Γ V := rowspan R ( V ) ∩ Z n . Here span R \operatorname{span}_{\mathbb R} span R Γ U \Gamma_U Γ U Γ V \Gamma_V Γ V d 1 d_1 d 1 n − d 2 n-d_2 n − d 2 U V T = 0 UV^{\mathsf T}=0 U V T = 0 Λ V : = Γ V ⊥ Z \Lambda_V := \Gamma_V^{\perp_{\mathbb Z}} Λ V := Γ V ⊥ Z Γ U ⊆ Λ V \Gamma_U \subseteq\Lambda_V Γ U ⊆ Λ V Λ V \Lambda_V Λ V d 2 d_2 d 2
Γ U ⊆ Λ V ⊆ Z n (103) \Gamma_U \subseteq\Lambda_V \subseteq\mathbb{Z}^{n} \tag*{(103)} Γ U ⊆ Λ V ⊆ Z n (103) is a flag of type ( d 1 , d 2 ) (d_1,d_2) ( d 1 , d 2 ) Z n \mathbb{Z}^n Z n det Λ V = det Γ V \det\Lambda_V = \det\Gamma_V det Λ V = det Γ V Γ V = Λ V ⊥ Z \Gamma_V = \Lambda_V^{\perp_{\mathbb Z}} Γ V = Λ V ⊥ Z ( Γ U , Γ V ) (\Gamma_U,\Gamma_V) ( Γ U , Γ V )
We next count the matrices giving a fixed pair of row lattices. Since U U U d 1 d_1 d 1 d 1 d_1 d 1 O n ( R ) O_n(R) O n ( R ) λ d 1 ( Γ U ) ≪ n R \lambda_{d_1}(\Gamma_U) \ll_n R λ d 1 ( Γ U ) ≪ n R d 1 d_1 d 1 O n ( R d 1 ) O_n(R^{d_1}) O n ( R d 1 ) Γ U \Gamma_U Γ U 1 ≤ det Γ U ≪ n R d 1 1 \leq\det\Gamma_U \ll_n R^{d_1} 1 ≤ det Γ U ≪ n R d 1
λ n − d 2 ( Γ V ) ≪ n R , 1 ≤ det Γ V ≪ n R n − d 2 . \lambda_{n-d_2}(\Gamma_V) \ll_n R,\qquad1 \leq\det\Gamma_V \ll_n R^{n-d_2}. λ n − d 2 ( Γ V ) ≪ n R , 1 ≤ det Γ V ≪ n R n − d 2 . It follows from (B.8) that, once Γ U \Gamma_U Γ U Γ V \Gamma_V Γ V
# { U } ≪ n , s ( R d 1 det Γ U ) s , # { V } ≪ n , t ( R n − d 2 det Γ V ) t . (104) \#\{U\} \ll_{n,s} \left(\frac{R^{d_1}}{\det\Gamma_U}\right)^s,\qquad
\#\{V\} \ll_{n,t} \left(\frac{R^{n-d_2}}{\det\Gamma_V}\right)^t. \tag*{(104)} # { U } ≪ n , s ( det Γ U R d 1 ) s , # { V } ≪ n , t ( det Γ V R n − d 2 ) t . (104) Let N d 1 , d 2 ( R ) N_{d_1,d_2}(R) N d 1 , d 2 ( R ) J 1 , J 2 ≥ 0 J_1,J_2 \geq0 J 1 , J 2 ≥ 0
2 j ≤ det Γ U < 2 j + 1 ( 0 ≤ j ≤ J 1 ) , 2 ℓ ≤ det Γ V < 2 ℓ + 1 ( 0 ≤ ℓ ≤ J 2 ) , 2^j \leq\det\Gamma_U < 2^{j+1}\quad(0 \leq j \leq J_1),\qquad
2^\ell\leq\det\Gamma_V < 2^{\ell+1}\quad(0 \leq\ell\leq J_2), 2 j ≤ det Γ U < 2 j + 1 ( 0 ≤ j ≤ J 1 ) , 2 ℓ ≤ det Γ V < 2 ℓ + 1 ( 0 ≤ ℓ ≤ J 2 ) , with
2 J 1 ≍ n R d 1 , 2 J 2 ≍ n R n − d 2 . 2^{J_1} \asymp_n R^{d_1},\qquad2^{J_2} \asymp_n R^{n-d_2}. 2 J 1 ≍ n R d 1 , 2 J 2 ≍ n R n − d 2 . For fixed j j j ℓ \ell ℓ 2 j + 1 2^{j+1} 2 j + 1 2 ℓ + 1 2^{\ell+1} 2 ℓ + 1 ( Γ U , Γ V ) (\Gamma_U,\Gamma_V) ( Γ U , Γ V )
≪ n , d 1 , d 2 2 j d 2 2 ℓ ( n − d 1 ) . \ll_{n,d_1,d_2} 2^{j d_2}2^{\ell(n-d_1)}. ≪ n , d 1 , d 2 2 j d 2 2 ℓ ( n − d 1 ) . For each such pair, (B.11) bounds the number of matrices by
≪ n , s , t R d 1 s + ( n − d 2 ) t 2 − j s 2 − ℓ t . \ll_{n,s,t} R^{d_1s+(n-d_2)t}2^{-js}2^{-\ell t}. ≪ n , s , t R d 1 s + ( n − d 2 ) t 2 − j s 2 − ℓ t . Thus
N d 1 , d 2 ( R ) ≪ n , s , t R d 1 s + ( n − d 2 ) t ∑ j = 0 J 1 ∑ ℓ = 0 J 2 2 j ( d 2 − s ) 2 ℓ ( n − d 1 − t ) = R d 1 s + ( n − d 2 ) t ( ∑ j = 0 J 1 2 j ( d 2 − s ) ) ( ∑ ℓ = 0 J 2 2 ℓ ( n − d 1 − t ) ) ≪ n , s , t R d 1 s + ( n − d 2 ) t 2 J 1 ( d 2 − s ) 2 J 2 ( n − d 1 − t ) ≪ n , s , t R d 1 s + ( n − d 2 ) t R d 1 ( d 2 − s ) R ( n − d 2 ) ( n − d 1 − t ) = R d 1 d 2 + ( n − d 2 ) ( n − d 1 ) = R n 2 − n ( d 1 + d 2 ) + 2 d 1 d 2 . \begin{aligned}
N_{d_1,d_2}(R)
&\ll_{n,s,t} R^{d_1s+(n-d_2)t}
\sum_{j=0}^{J_1}\sum_{\ell=0}^{J_2}
2^{j(d_2-s)}2^{\ell(n-d_1-t)} \\
&= R^{d_1s+(n-d_2)t}
\left(\sum_{j=0}^{J_1}2^{j(d_2-s)}\right)
\left(\sum_{\ell=0}^{J_2}2^{\ell(n-d_1-t)}\right) \\
&\ll_{n,s,t} R^{d_1s+(n-d_2)t}
2^{J_1(d_2-s)}2^{J_2(n-d_1-t)} \\
&\ll_{n,s,t} R^{d_1s+(n-d_2)t}R^{d_1(d_2-s)}R^{(n-d_2)(n-d_1-t)} \\
&= R^{d_1d_2+(n-d_2)(n-d_1)} \\
&= R^{n^2-n(d_1+d_2)+2d_1d_2}.
\end{aligned} N d 1 , d 2 ( R ) ≪ n , s , t R d 1 s + ( n − d 2 ) t j = 0 ∑ J 1 ℓ = 0 ∑ J 2 2 j ( d 2 − s ) 2 ℓ ( n − d 1 − t ) = R d 1 s + ( n − d 2 ) t ( j = 0 ∑ J 1 2 j ( d 2 − s ) ) ( ℓ = 0 ∑ J 2 2 ℓ ( n − d 1 − t ) ) ≪ n , s , t R d 1 s + ( n − d 2 ) t 2 J 1 ( d 2 − s ) 2 J 2 ( n − d 1 − t ) ≪ n , s , t R d 1 s + ( n − d 2 ) t R d 1 ( d 2 − s ) R ( n − d 2 ) ( n − d 1 − t ) = R d 1 d 2 + ( n − d 2 ) ( n − d 1 ) = R n 2 − n ( d 1 + d 2 ) + 2 d 1 d 2 . It remains to maximize the exponent
E ( d 1 , d 2 ) : = n 2 − n ( d 1 + d 2 ) + 2 d 1 d 2 E(d_1,d_2) := n^2 - n(d_1+d_2) + 2d_1d_2 E ( d 1 , d 2 ) := n 2 − n ( d 1 + d 2 ) + 2 d 1 d 2 over 1 ≤ d 1 ≤ s 1 \le d_1 \le s 1 ≤ d 1 ≤ s n − t ≤ d 2 ≤ n − 1 n-t \le d_2 \le n-1 n − t ≤ d 2 ≤ n − 1
E ( d 1 + 1 , d 2 ) − E ( d 1 , d 2 ) = 2 d 2 − n ≥ n − 2 t ≥ 0 E(d_1+1,d_2)-E(d_1,d_2)=2d_2-n \ge n-2t \ge0 E ( d 1 + 1 , d 2 ) − E ( d 1 , d 2 ) = 2 d 2 − n ≥ n − 2 t ≥ 0 and
E ( d 1 , d 2 − 1 ) − E ( d 1 , d 2 ) = n − 2 d 1 ≥ n − 2 s ≥ 0. E(d_1,d_2-1)-E(d_1,d_2)=n-2d_1 \ge n-2s \ge0. E ( d 1 , d 2 − 1 ) − E ( d 1 , d 2 ) = n − 2 d 1 ≥ n − 2 s ≥ 0. Thus E ( d 1 , d 2 ) E(d_1,d_2) E ( d 1 , d 2 ) d 1 = s d_1=s d 1 = s d 2 = n − t d_2=n-t d 2 = n − t
E ( s , n − t ) = n ( s + t ) − 2 s t . E(s,n-t)=n(s+t)-2st. E ( s , n − t ) = n ( s + t ) − 2 s t . Summing over the O n ( 1 ) O_n(1) O n ( 1 ) ( d 1 , d 2 ) (d_1,d_2) ( d 1 , d 2 ) □ \square □
