Figure 10.
the size of the weights and on a concentration bound for signed subset sums. Finally, the solver is read inside a general theory of constraint propagation, in which every network factors as T = T c o m p ∘ T s t r u c t T = T_{\mathrm{comp}} \circ T_{\mathrm{struct}} T = T comp ∘ T struct
Introduction
Two questions can be asked about the subset sums of a set of positive integers: whether a target is reached, and how many times. For the primes the answers live at very different depths.
The floor. Let W N = { 1 } ∪ { p ≤ N } W_N = \{1\} \cup\{p \le N\} W N = { 1 } ∪ { p ≤ N } p k + 1 < 2 p k p_{k+1} < 2p_k p k + 1 < 2 p k § 3 ). For the primes the decision problem is therefore trivial, and it rests on the prime number theorem at its weakest.
The ceiling. The number r N ( T ) r_N(T) r N ( T ) T T T z T z^T z T F N ( z ) = ∏ w ∈ W N ( 1 + z w ) F_N(z) = \prod_{w \in W_N}(1+z^w) F N ( z ) = ∏ w ∈ W N ( 1 + z w ) z = 1 z = 1 z = 1 r N ( T ) r_N(T) r N ( T ) § 4.8 ). Away from z = 1 z = 1 z = 1 a / q a/q a / q 2 ∣ cos ( π w a / q ) ∣ 2|\cos(\pi wa/q)| 2∣ cos ( π w a / q ) ∣ log ∣ F N ( e ( a / q ) ) ∣ \log|F_N(e(a/q))| log ∣ F N ( e ( a / q )) ∣ p p p q q q
The thread. The paper follows one object, the arc, through three settings.
In §2–3 the weights are the primes. We determine which arcs rise above the sea of the minor arcs (Lemma 1 ), bound the neighbourhoods of all arcs uniformly in q q q Theorem E ).
In §4 we read the height of an arc as a prime race. Under two conditions on the characters it sees, ( R ) (R) ( R ) ( N ) (N) ( N ) Proposition 3 ); both conditions are verified for q = 8 , 10 , 14 q = 8, 10, 14 q = 8 , 10 , 14 ζ \zeta ζ Proposition 30 ). They are reformulations of the Knapowski–Turán type, in which the hypothesis becomes a statement about counting subsets.
In §5–6 the weights are arbitrary. The floor fails, the dummies of HURS restore it, and the arcs, once the dummies are removed exactly, which makes them unnecessary in the solver, become the decision rule of a Subset Sum solver. We give its cost law, its benchmarks, and the rigorous form of its guarantees on random and on dense instances.
The arc, seen from far enough, is propagation on a quotient. In §7 we develop the general theory of constraint propagation behind the algorithm: a structural part (equalities with parity, functional definitions on a complete DAG) solved in one pass, a computational part iterated to a fixpoint, the convergence of the full operator, and definitional variables that make implicit links evaluable. In §8 we read the solver of §5 inside it.
Why it matters. Prime races and Subset Sum have so far been separate subjects. The first belongs to analytic number theory (Chebyshev’s bias, Knapowski–Turán, Rubinstein–Sarnak); the second is a classical NP-complete problem whose algorithms are combinatorial. The arcs join them. Counting subsets of the primes is a circle-method problem in which every arc is a prime race, so a hypothesis about the zeros of Dirichlet L L L q = 2 k q = 2^k q = 2 k 2 p j 2p^j 2 p j Λ ( q / 2 ) / ϕ ( q ) \Lambda(q/2)/\phi(q) Λ ( q /2 ) / ϕ ( q )
§9 places the results in the literature and states their scope.
Notation. e ( x ) = e 2 π i x e(x) = e^{2\pi i x} e ( x ) = e 2 π i x Λ \Lambda Λ ϕ \phi ϕ Φ q \Phi_q Φ q q q q π ( N , χ ) = ∑ p ≤ N χ ( p ) \pi(N,\chi) = \sum_{p\le N}\chi(p) π ( N , χ ) = ∑ p ≤ N χ ( p ) li x = ∫ 2 x d t log t + li 2 \operatorname{li}x = \int_2^x \frac{dt}{\log t}+\operatorname{li}2 li x = ∫ 2 x l o g t d t + li 2 C C C
Map of the results.
Table 12.
The floor: completeness and linear search
Define the omicron of a set of positive integers w 1 ≤ w 2 ≤ ⋯ ≤ w n w_1 \le w_2 \le\cdots\le w_n w 1 ≤ w 2 ≤ ⋯ ≤ w n o = max k w k + 1 / ( 1 + ∑ i ≤ k w i ) o = \max_k w_{k+1}/(1+\sum_{i\le k} w_i) o = max k w k + 1 / ( 1 + ∑ i ≤ k w i )
Lemma 2.1 (gap filling; Brown). Let w 1 ≤ w 2 ≤ ⋯ ≤ w n w_1 \le w_2 \le\cdots\le w_n w 1 ≤ w 2 ≤ ⋯ ≤ w n S k = ∑ i ≤ k w i S_k = \sum_{i\le k}w_i S k = ∑ i ≤ k w i S 0 = 0 S_0 = 0 S 0 = 0 0 ≤ T ≤ S n 0 \le T \le S_n 0 ≤ T ≤ S n w k + 1 ≤ 1 + S k w_{k+1} \le1+S_k w k + 1 ≤ 1 + S k 0 ≤ k < n 0 \le k < n 0 ≤ k < n o ≤ 1 o \le1 o ≤ 1 o ≥ w 1 ≥ 1 o \ge w_1 \ge1 o ≥ w 1 ≥ 1 o = 1 o = 1 o = 1 W N W_N W N
Proof. If the condition holds, the subset sums of the first k k k [ 0 , S k ] [0,S_k] [ 0 , S k ] w k + 1 ≤ S k + 1 w_{k+1} \le S_k+1 w k + 1 ≤ S k + 1 [ 0 , S k + 1 ] [0,S_{k+1}] [ 0 , S k + 1 ] w k + 1 > 1 + S k w_{k+1} > 1+S_k w k + 1 > 1 + S k k k k S k + 1 ≤ S n S_k+1 \le S_n S k + 1 ≤ S n k k k S k S_k S k w j ≥ w k + 1 > S k + 1 w_j \ge w_{k+1} > S_k+1 w j ≥ w k + 1 > S k + 1
Linear search. Scan the weights downwards and take w k w_k w k w k ≤ T r e m w_k \le T_{\mathrm{rem}} w k ≤ T rem w k w_k w k T r e m ≤ ∑ i ≤ k w i T_{\mathrm{rem}} \le\sum_{i\le k} w_i T rem ≤ ∑ i ≤ k w i T r e m ≥ w k T_{\mathrm{rem}} \ge w_k T rem ≥ w k T r e m − w k ≤ ∑ i < k w i T_{\mathrm{rem}}-w_k \le\sum_{i<k}w_i T rem − w k ≤ ∑ i < k w i T r e m ≤ w k − 1 ≤ ∑ i < k w i T_{\mathrm{rem}} \le w_k-1 \le\sum_{i<k}w_i T rem ≤ w k − 1 ≤ ∑ i < k w i T r e m = 0 T_{\mathrm{rem}}=0 T rem = 0 O ( n ) O(n) O ( n ) r N ( T ) ≥ 1 r_N(T) \ge1 r N ( T ) ≥ 1 0 ≤ T ≤ S N 0 \le T \le S_N 0 ≤ T ≤ S N r N ( T ) = r N ( S N − T ) r_N(T) = r_N(S_N-T) r N ( T ) = r N ( S N − T )
Proposition 2.3 (the primes without the weight 1). Let P N = { p ≤ N } P_N=\{p\le N\} P N = { p ≤ N } Σ N = ∑ p ≤ N p = S N − 1 \Sigma_N=\sum_{p\le N}p=S_N-1 Σ N = ∑ p ≤ N p = S N − 1 N ≥ 5 N\ge5 N ≥ 5 P N P_N P N
[ 0 , Σ N ] ∖ { 1 , 4 , 6 , Σ N − 6 , Σ N − 4 , Σ N − 1 } ; [0,\Sigma_N]\setminus\{1,4,6,\Sigma_N-6,\Sigma_N-4,\Sigma_N-1\}; [ 0 , Σ N ] ∖ { 1 , 4 , 6 , Σ N − 6 , Σ N − 4 , Σ N − 1 } ; for N = 2 N=2 N = 2 N = 3 N=3 N = 3 [ 0 , Σ N ] [0,\Sigma_N] [ 0 , Σ N ] { 1 } \{1\} { 1 } { 1 , 4 } \{1,4\} { 1 , 4 } N ≥ 7 N\ge7 N ≥ 7 { 0 } \{0\} { 0 } [ 2 , 3 ] [2,3] [ 2 , 3 ] { 5 } \{5\} { 5 } [ 7 , Σ N − 7 ] [7,\Sigma_N-7] [ 7 , Σ N − 7 ] { Σ N − 5 } \{\Sigma_N-5\} { Σ N − 5 } [ Σ N − 3 , Σ N − 2 ] [\Sigma_N-3,\Sigma_N-2] [ Σ N − 3 , Σ N − 2 ] { Σ N } \{\Sigma_N\} { Σ N }
Proof. For N = 5 , 7 , 11 N=5,7,11 N = 5 , 7 , 11 Σ = 10 , 17 , 28 \Sigma=10,17,28 Σ = 10 , 17 , 28 p n p_n p n Σ n = ∑ k ≤ n p k \Sigma_n=\sum_{k\le n}p_k Σ n = ∑ k ≤ n p k H = { 1 , 4 , 6 } H=\{1,4,6\} H = { 1 , 4 , 6 } R n R_n R n { p 1 , … , p n } \{p_1,\ldots,p_n\} { p 1 , … , p n } R n = [ 0 , Σ n ] ∖ ( H ∪ ( Σ n − H ) ) R_n=[0,\Sigma_n]\setminus(H\cup(\Sigma_n-H)) R n = [ 0 , Σ n ] ∖ ( H ∪ ( Σ n − H )) 7 ≤ p = p n + 1 ≤ Σ n − 13 7\le p=p_{n+1}\le\Sigma_n-13 7 ≤ p = p n + 1 ≤ Σ n − 13 R n + 1 = R n ∪ ( R n + p ) R_{n+1}=R_n\cup(R_n+p) R n + 1 = R n ∪ ( R n + p )
below 7 only R n R_n R n R n + p ≥ p ≥ 7 R_n+p\ge p\ge7 R n + p ≥ p ≥ 7
for j ∈ H j\in H j ∈ H Σ n − j − p \Sigma_n-j-p Σ n − j − p [ 7 , Σ n − 7 ] [7,\Sigma_n-7] [ 7 , Σ n − 7 ] 7 − j ≤ p ≤ Σ n − j − 7 7-j\le p\le\Sigma_n-j-7 7 − j ≤ p ≤ Σ n − j − 7 Σ n − j ∈ R n + p \Sigma_n-j\in R_n+p Σ n − j ∈ R n + p R n R_n R n
for Σ n < x ≤ Σ n + p \Sigma_n<x\le\Sigma_n+p Σ n < x ≤ Σ n + p x ∈ R n + 1 x\in R_{n+1} x ∈ R n + 1 y = x − p ∈ R n y=x-p\in R_n y = x − p ∈ R n y > Σ n − p ≥ 13 y>\Sigma_n-p\ge13 y > Σ n − p ≥ 13 y ∉ H y\notin H y ∈ / H y ∉ R n y\notin R_n y ∈ / R n y ∈ Σ n − H y\in\Sigma_n-H y ∈ Σ n − H x ∈ Σ n + 1 − H x\in\Sigma_{n+1}-H x ∈ Σ n + 1 − H
Hence R n + 1 = [ 0 , Σ n + 1 ] ∖ ( H ∪ ( Σ n + 1 − H ) ) R_{n+1}=[0,\Sigma_{n+1}]\setminus\left(H\cup(\Sigma_{n+1}-H)\right) R n + 1 = [ 0 , Σ n + 1 ] ∖ ( H ∪ ( Σ n + 1 − H ) ) p p p n ≥ 5 n\ge5 n ≥ 5 D n = Σ n − 1 − p n D_n=\Sigma_{n-1}-p_n D n = Σ n − 1 − p n D n + 1 = Σ n − p n + 1 > Σ n − 2 p n = D n D_{n+1}=\Sigma_n-p_{n+1}>\Sigma_n-2p_n=D_n D n + 1 = Σ n − p n + 1 > Σ n − 2 p n = D n D D D D 6 = Σ 5 − p 6 = 28 − 13 = 15 ≥ 13 D_6=\Sigma_5-p_6=28-13=15\ge13 D 6 = Σ 5 − p 6 = 28 − 13 = 15 ≥ 13 p n + 1 ≥ 13 p_{n+1}\ge13 p n + 1 ≥ 13 N = 11 N=11 N = 11 N N N
The proposition was also checked directly for every N < 2 ⋅ 10 4 N<2\cdot10^4 N < 2 ⋅ 1 0 4 N N N Σ N ≥ m + 7 \Sigma_N\ge m+7 Σ N ≥ m + 7 W N W_N W N 4 = 3 + 1 4=3+1 4 = 3 + 1 6 = 5 + 1 6=5+1 6 = 5 + 1
Remark 2.4 (interval representation). The reachable sets obey R k + 1 = R k ∪ ( R k + w k + 1 ) R_{k+1}=R_k\cup(R_k+w_{k+1}) R k + 1 = R k ∪ ( R k + w k + 1 ) Z \mathbb{Z} Z A + B A+B A + B B [ Z ] \mathbb{B}[\mathbb{Z}] B [ Z ] ∅ \varnothing ∅ { 0 } \{0\} { 0 } w w w { w } \{w\} { w } R n = ∏ k ( { 0 } ⊕ { w k } ) R_n=\prod_k(\{0\}\oplus\{w_k\}) R n = ∏ k ({ 0 } ⊕ { w k }) F W ( z ) = ∏ k ( 1 + z w k ) F_W(z)=\prod_k(1+z^{w_k}) F W ( z ) = ∏ k ( 1 + z w k ) ⊕ \oplus ⊕ R k R_k R k K k K_k K k O ( K k ) O(K_k) O ( K k ) K k ≤ min ( 2 k , S k + 1 ) K_k\le\min(2^k,S_k+1) K k ≤ min ( 2 k , S k + 1 ) K = 1 K=1 K = 1 K = 7 K=7 K = 7 N ≥ 7 N\ge7 N ≥ 7 K = 1 K=1 K = 1
The floor is blind to the zeros of ζ \zeta ζ F N F_N F N
The arcs of the prime product
Factors that vanish, and the Fourier coefficients of the count
For a rational a / q a/q a / q ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 q ≥ 3 q\ge3 q ≥ 3 F N ( e ( a / q ) ) F_N(e(a/q)) F N ( e ( a / q )) 1 + e ( w a / q ) 1+e(wa/q) 1 + e ( w a / q ) 2 ∣ cos ( π w a / q ) ∣ 2\lvert\cos(\pi wa/q)\rvert 2 ∣ cos ( π w a / q )∣ w ≡ q / 2 ( m o d q ) w\equiv q/2\pmod q w ≡ q /2 ( mod q ) q ≥ 3 q\ge3 q ≥ 3 w ∈ W N w\in W_N w ∈ W N q q q w = q / 2 w=q/2 w = q /2 q / 2 q/2 q /2 q / 2 q/2 q /2 q = 4 q=4 q = 4 w = 2 w=2 w = 2 q = 2 p q=2p q = 2 p w = p w=p w = p V ( a / q ) \mathcal{V}(a/q) V ( a / q ) F N ∗ ( a / q ) F_N^*(a/q) F N ∗ ( a / q ) q = 2 q=2 q = 2
Remark 3.1 (the amplitudes are Fourier coefficients of the count). Since F N F_N F N
F N ( e ( a / q ) ) = ∑ T = 0 S N r N ( T ) e ( T a / q ) F_N(e(a/q))=\sum_{T=0}^{S_N}r_N(T)e(Ta/q) F N ( e ( a / q )) = T = 0 ∑ S N r N ( T ) e ( T a / q ) exactly.
If V ( a / q ) = { w } \mathcal{V}(a/q)=\{w\} V ( a / q ) = { w } w = q / 2 w=q/2 w = q /2 F N ( e ( a / q ) ) = 0 F_N(e(a/q))=0 F N ( e ( a / q )) = 0 F N = ( 1 + z w ) F N ∗ F_N=(1+z^w)F_N^* F N = ( 1 + z w ) F N ∗ ζ = e ( a / q ) \zeta=e(a/q) ζ = e ( a / q ) ζ w = − 1 \zeta^w=-1 ζ w = − 1 ζ F N ′ ( ζ ) = − w F N ∗ ( ζ ) \zeta F_N'(\zeta)=-wF_N^*(\zeta) ζ F N ′ ( ζ ) = − w F N ∗ ( ζ )
F N ∗ ( a / q ) = − 1 w ∑ T T r N ( T ) e ( T a / q ) . F_N^*(a/q)=-\frac{1}{w}\sum_T T r_N(T)e(Ta/q). F N ∗ ( a / q ) = − w 1 T ∑ T r N ( T ) e ( T a / q ) . Equivalently, F N ∗ ( a / q ) F_N^*(a/q) F N ∗ ( a / q ) W N ∖ { w } W_N\setminus\{w\} W N ∖ { w }
So every arc amplitude is an exact Fourier coefficient: of r N ( T ) r_N(T) r N ( T ) T ⋅ r N ( T ) T\cdot r_N(T) T ⋅ r N ( T ) T T T q q q q q q
Which arcs emerge
Lemma 1. For q ≥ 3 q\ge3 q ≥ 3 ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 w ˉ ( q ) = φ ( q ) − 1 ∑ ( r , q ) = 1 log ∣ cos ( π r a / q ) ∣ \bar{w}(q)=\varphi(q)^{-1}\sum_{(r,q)=1}\log|\cos(\pi ra/q)| w ˉ ( q ) = φ ( q ) − 1 ∑ ( r , q ) = 1 log ∣ cos ( π r a / q ) ∣ w ˉ ( q ) \bar{w}(q) w ˉ ( q ) a a a
w ˉ ( q ) = { − log 2 , q odd , − log 2 + Λ ( q / 2 ) / φ ( q ) , q even . \bar{w}(q)=
\begin{cases}
-\log2, & q\ \text{odd},\\
-\log2+\Lambda(q/2)/\varphi(q), & q\ \text{even}.
\end{cases} w ˉ ( q ) = { − log 2 , − log 2 + Λ ( q /2 ) / φ ( q ) , q odd , q even . Proof. Multiplication by a a a q q q ∣ 1 + e ( r / q ) ∣ = 2 ∣ cos ( π r / q ) ∣ |1+e(r/q)|=2|\cos(\pi r/q)| ∣1 + e ( r / q ) ∣ = 2∣ cos ( π r / q ) ∣ ∑ ( r , q ) = 1 log ∣ 2 cos ( π r / q ) ∣ = log ∣ ∏ ( 1 + ζ q r ) ∣ = log ∣ Φ q ( − 1 ) ∣ \sum_{(r,q)=1}\log|2\cos(\pi r/q)|=\log|\prod(1+\zeta_q^r)|=\log|\Phi_q(-1)| ∑ ( r , q ) = 1 log ∣2 cos ( π r / q ) ∣ = log ∣ ∏ ( 1 + ζ q r ) ∣ = log ∣ Φ q ( − 1 ) ∣ q > 2 q>2 q > 2 Φ q ( − 1 ) = p \Phi_q(-1)=p Φ q ( − 1 ) = p q = 2 p j q=2p^j q = 2 p j p p p p = 2 p=2 p = 2 Φ q ( − 1 ) = 1 \Phi_q(-1)=1 Φ q ( − 1 ) = 1 log ∣ Φ q ( − 1 ) ∣ = Λ ( q / 2 ) \log|\Phi_q(-1)|=\Lambda(q/2) log ∣ Φ q ( − 1 ) ∣ = Λ ( q /2 ) q q q 0 0 0 q q q
The lemma rests on classical facts about cyclotomic values; the reading as a level of the arcs is ours. It was checked against direct computation for every 3 ≤ q < 400 3\le q<400 3 ≤ q < 400
The sea. For almost every θ \theta θ ( 1 / π ( N ) ) ∑ p ≤ N log ∣ cos ( π p θ ) ∣ → ∫ 0 1 log ∣ cos π x ∣ d x = − log 2 (1/\pi(N))\sum_{p\le N}\log|\cos(\pi p\theta)|\to\int_0^1\log|\cos\pi x|\,dx=-\log2 ( 1/ π ( N )) ∑ p ≤ N log ∣ cos ( π pθ ) ∣ → ∫ 0 1 log ∣ cos π x ∣ d x = − log 2 p θ p\theta pθ a / q a/q a / q w ˉ ( q ) > − log 2 \bar{w}(q)>-\log2 w ˉ ( q ) > − log 2 q = 2 k q=2^k q = 2 k k ≥ 2 k\ge2 k ≥ 2 q = 2 p j q=2p^j q = 2 p j p p p q q q q = 1 q=1 q = 1 emerging moduli , E \mathcal{E} E q / 2 q/2 q /2
Up to O q ( 1 ) O_q(1) O q ( 1 ) p ∣ q p\mid q p ∣ q
log ∣ F N ∗ ( a / q ) ∣ = π ( N ) ( log 2 + w ˉ ( q ) ) + H a / q ( N ) + O q ( 1 ) , H a / q ( N ) = ∑ p ≤ N p ∤ q c p ( m o d q ) , \log|F_N^*(a/q)|=\pi(N)\bigl(\log2+\bar{w}(q)\bigr)+H_{a/q}(N)+O_q(1),\qquad H_{a/q}(N)=\sum_{\substack{p\le N\\p\nmid q}}c_p\pmod q, log ∣ F N ∗ ( a / q ) ∣ = π ( N ) ( log 2 + w ˉ ( q ) ) + H a / q ( N ) + O q ( 1 ) , H a / q ( N ) = p ≤ N p ∤ q ∑ c p ( mod q ) , with c r = log ∣ cos ( π r a / q ) ∣ − w ˉ ( q ) c_r=\log|\cos(\pi ra/q)|-\bar{w}(q) c r = log ∣ cos ( π r a / q ) ∣ − w ˉ ( q ) H a / q H_{a/q} H a / q q q q c r c_r c r ∑ r c r = 0 \sum_r c_r=0 ∑ r c r = 0
Figure 1 (PDF p. 7) shows the levels at N = 2000 N = 2000 N = 2000 q ∈ E q \in\mathcal{E} q ∈ E
Figure 1 (PDF p. 7). The arcs of the prime product at N = 2000 N = 2000 N = 2000 ( 1 / ∣ W N ∣ ) ∑ w log ∣ 2 cos ( π w θ ) ∣ (1/|W_N|)\sum_w \log|2\cos(\pi w\theta)| ( 1/∣ W N ∣ ) ∑ w log ∣2 cos ( π w θ ) ∣ θ ∈ ( 0 , 1 2 ] \theta\in(0,\frac{1}{2}] θ ∈ ( 0 , 2 1 ] θ = a / q \theta= a/q θ = a / q q ≤ 40 q \le40 q ≤ 40 q = 2 k q = 2^k q = 2 k 2 p j 2p^j 2 p j Λ ( q / 2 ) / φ ( q ) \Lambda(q/2)/\varphi(q) Λ ( q /2 ) / φ ( q ) O q ( 1 ) O_q(1) O q ( 1 )
Neighbourhoods of the arcs: the lobes
Near an arc the level changes. For q ≥ 1 q \ge1 q ≥ 1 X ≥ 0 X \ge0 X ≥ 0
A q ( X ) = 1 X ∫ 0 X log ∣ Φ q ( − e ( x ) ) ∣ d x ( A q ( 0 ) = log ∣ Φ q ( − 1 ) ∣ ) . A_q(X) = \frac{1}{X}\int_0^X \log|\Phi_q(-e(x))|\,dx \qquad(A_q(0) = \log|\Phi_q(-1)|). A q ( X ) = X 1 ∫ 0 X log ∣ Φ q ( − e ( x )) ∣ d x ( A q ( 0 ) = log ∣ Φ q ( − 1 ) ∣ ) . By the identity ∏ ( r , q ) = 1 ( 1 + ζ q r y ) = Φ q ( − y ) \prod_{(r,q)=1}(1+\zeta_q^r y)=\Phi_q(-y) ∏ ( r , q ) = 1 ( 1 + ζ q r y ) = Φ q ( − y ) A q ( X ) / φ ( q ) A_q(X)/\varphi(q) A q ( X ) / φ ( q ) ∑ p log ∣ 2 cos ( π ( p a / q + p β ) ) ∣ \sum_p \log|2\cos(\pi(pa/q+p\beta))| ∑ p log ∣2 cos ( π ( p a / q + pβ )) ∣ β = X / N \beta= X/N β = X / N core level is c q = A q ( 0 ) / φ ( q ) c_q=A_q(0)/\varphi(q) c q = A q ( 0 ) / φ ( q ) log 2 \log2 log 2 q = 1 q=1 q = 1 Λ ( q / 2 ) / φ ( q ) \Lambda(q/2)/\varphi(q) Λ ( q /2 ) / φ ( q ) q q q q q q ≥ 3 \ge3 ≥ 3 ℓ q = sup { A q ( X ) / φ ( q ) : X ≥ X q } \ell_q=\sup\{A_q(X)/\varphi(q):X\ge X_q\} ℓ q = sup { A q ( X ) / φ ( q ) : X ≥ X q } X q X_q X q A q A_q A q A q A_q A q ℓ q ∗ = sup { A q ( X ) / φ ( q ) : X ≥ 1 / ( 2 q ) } \ell_q^*=\sup\{A_q(X)/\varphi(q):X\ge1/(2q)\} ℓ q ∗ = sup { A q ( X ) / φ ( q ) : X ≥ 1/ ( 2 q )} X q X_q X q
Table 17.
Table 1 (PDF p. 8). Core and lobe levels (exact integrals of log ∣ Φ q ∣ \log|\Phi_q| log ∣ Φ q ∣
Figure 2. The neighbourhood of an arc. Blue: the per-prime excess at θ = a / q + X / N \theta= a/q + X/N θ = a / q + X / N N = 2 ⋅ 10 5 N = 2 \cdot10^5 N = 2 ⋅ 1 0 5 A q ( X ) / φ ( q ) A_q(X)/\varphi(q) A q ( X ) / φ ( q ) § 4.6 ). (a) The emerging arc 1 / 6 1/6 1/6 0.549 0.549 0.549 1 / 3 1/3 1/3 0 0 0 ℓ 3 = 0.1316 \ell_3 = 0.1316 ℓ 3 = 0.1316 X ≈ 0.73 X \approx0.73 X ≈ 0.73
Lemma P (maximum principle for emerging arcs). For every emerging q ≥ 4 q \ge4 q ≥ 4 x ↦ ∣ Φ q ( − e ( x ) ) ∣ x \mapsto|\Phi_q(-e(x))| x ↦ ∣ Φ q ( − e ( x )) ∣ Φ q ( − 1 ) = e Λ ( q / 2 ) \Phi_q(-1) = e^{\Lambda(q/2)} Φ q ( − 1 ) = e Λ ( q /2 ) x = 0 x = 0 x = 0 A q ( X ) < Λ ( q / 2 ) A_q(X) < \Lambda(q/2) A q ( X ) < Λ ( q /2 ) X > 0 X > 0 X > 0 ℓ q < c q \ell_q < c_q ℓ q < c q
Proof. With y = − e ( x ) y = -e(x) y = − e ( x ) Φ q ( z ) = ∏ d ∣ q ( z d − 1 ) μ ( q / d ) \Phi_q(z) = \prod_{d\mid q}(z^d - 1)^{\mu(q/d)} Φ q ( z ) = ∏ d ∣ q ( z d − 1 ) μ ( q / d )
q = 2 k q = 2^k q = 2 k k ≥ 2 k \ge2 k ≥ 2 Φ q ( z ) = z q / 2 + 1 \Phi_q(z) = z^{q/2} + 1 Φ q ( z ) = z q /2 + 1 ∣ Φ q ( y ) ∣ = 2 ∣ cos ( π q x / 2 ) ∣ ≤ 2 |\Phi_q(y)| = 2|\cos(\pi qx/2)| \le2 ∣ Φ q ( y ) ∣ = 2∣ cos ( π q x /2 ) ∣ ≤ 2
q = 2 p j q = 2p^j q = 2 p j p p p d ∈ { q , q / 2 , q / p , q / ( 2 p ) } d \in\{q,q/2,q/p,q/(2p)\} d ∈ { q , q /2 , q / p , q / ( 2 p )} ∣ Φ q ( − e ( x ) ) ∣ = ∣ sin p θ / sin θ ∣ |\Phi_q(-e(x))| = |\sin p\theta/\sin\theta| ∣ Φ q ( − e ( x )) ∣ = ∣ sin pθ / sin θ ∣ θ = π p j − 1 x \theta= \pi p^{j-1}x θ = π p j − 1 x p p p
In both cases log ∣ Φ q ( − e ( x ) ) ∣ ≤ Λ ( q / 2 ) \log|\Phi_q(-e(x))| \le\Lambda(q/2) log ∣ Φ q ( − e ( x )) ∣ ≤ Λ ( q /2 ) 0 0 0 A q ( X ) → 0 A_q(X) \to0 A q ( X ) → 0 X ≥ X q X \ge X_q X ≥ X q Λ ( q / 2 ) \Lambda(q/2) Λ ( q /2 )
Lemma Q (lobes of every arc). Let q ≥ 3 q \ge3 q ≥ 3 h q ( x ) = log ∣ Φ q ( − e ( x ) ) ∣ h_q(x) = \log|\Phi_q(-e(x))| h q ( x ) = log ∣ Φ q ( − e ( x )) ∣ κ = Cl 2 ( π / 3 ) / π = 0.32307 … \kappa= \operatorname{Cl}_2(\pi/3)/\pi= 0.32307\ldots κ = Cl 2 ( π /3 ) / π = 0.32307 …
h q h_q h q [ 0 , 1 / ( 2 q ) ) [0,1/(2q)) [ 0 , 1/ ( 2 q )) A q A_q A q ( 0 , 1 / ( 2 q ) ) (0,1/(2q)) ( 0 , 1/ ( 2 q )) A q ( X ) < φ ( q ) c q A_q(X) < \varphi(q)c_q A q ( X ) < φ ( q ) c q 0 < X < 1 / ( 2 q ) 0 < X < 1/(2q) 0 < X < 1/ ( 2 q ) X q ≥ 1 / ( 2 q ) X_q \ge1/(2q) X q ≥ 1/ ( 2 q )
For every X ≥ 1 / ( 2 q ) X \ge1/(2q) X ≥ 1/ ( 2 q )
A q ( X ) ≤ ∑ μ ( q / d ) = 1 min ( log 2 , κ d X ) + ∑ μ ( q / d ) = − 1 β ( d X ) , β ( L ) = { max ( κ , min ( κ / L , log e 2 L ) ) L ≤ 1 , κ / L L > 1 , A_q(X) \le\sum_{\mu(q/d)=1} \min\left(\log2,\frac{\kappa}{dX}\right)+\sum_{\mu(q/d)=-1}\beta(dX),\qquad
\beta(L)=
\begin{cases}
\max\left(\kappa,\min\left(\kappa/L,\log\frac{e}{2L}\right)\right) & L\le1,\\
\kappa/L & L>1,
\end{cases} A q ( X ) ≤ μ ( q / d ) = 1 ∑ min ( log 2 , d X κ ) + μ ( q / d ) = − 1 ∑ β ( d X ) , β ( L ) = { max ( κ , min ( κ / L , log 2 L e ) ) κ / L L ≤ 1 , L > 1 , and the right side is non-increasing in X X X R q R_q R q X = 1 / ( 2 q ) X=1/(2q) X = 1/ ( 2 q ) φ ( q ) \varphi(q) φ ( q )
ℓ q ≤ R q ≤ 2 ω ( q ) − 1 ( 1 + log 2 + log rad q ) φ ( q ) ≤ τ ( q ) log ( 2 e q ) 2 φ ( q ) . \ell_q \le R_q \le\frac{2^{\omega(q)-1}(1+\log2+\log\operatorname{rad}q)}{\varphi(q)} \le\frac{\tau(q)\log(2eq)}{2\varphi(q)}. ℓ q ≤ R q ≤ φ ( q ) 2 ω ( q ) − 1 ( 1 + log 2 + log rad q ) ≤ 2 φ ( q ) τ ( q ) log ( 2 e q ) . For X ≥ 1 X\ge1 X ≥ 1 A q ( X ) / φ ( q ) ≤ κ τ ( q ) / φ ( q ) A_q(X)/\varphi(q)\le\kappa\tau(q)/\varphi(q) A q ( X ) / φ ( q ) ≤ κ τ ( q ) / φ ( q )
Proof.
(1) Pairing r r r q − r q-r q − r h q ( x ) = ∑ log ∣ c r + 2 cos 2 π x ∣ h_q(x)=\sum\log|c_r+2\cos2\pi x| h q ( x ) = ∑ log ∣ c r + 2 cos 2 π x ∣ φ ( q ) / 2 \varphi(q)/2 φ ( q ) /2 c r = 2 cos ( 2 π r / q ) ≥ − 2 cos ( π / q ) c_r=2\cos(2\pi r/q)\ge-2\cos(\pi/q) c r = 2 cos ( 2 π r / q ) ≥ − 2 cos ( π / q ) ∣ 2 r / q − 1 ∣ ≥ 1 / q |2r/q-1|\ge1/q ∣2 r / q − 1∣ ≥ 1/ q [ 0 , 1 / ( 2 q ) ) [0,1/(2q)) [ 0 , 1/ ( 2 q ))
(2) By Möbius inversion h q ( x ) = ∑ d ∣ q μ ( q / d ) G ( d x + d / 2 ) h_q(x)=\sum_{d\mid q}\mu(q/d)G(dx+d/2) h q ( x ) = ∑ d ∣ q μ ( q / d ) G ( d x + d /2 ) G ( u ) = log ∣ 2 sin π u ∣ G(u)=\log|2\sin\pi u| G ( u ) = log ∣2 sin π u ∣ ∫ 0 u G = − Cl 2 ( 2 π u ) / ( 2 π ) \int_0^u G=-\operatorname{Cl}_2(2\pi u)/(2\pi) ∫ 0 u G = − Cl 2 ( 2 π u ) / ( 2 π ) ∣ Cl 2 ∣ ≤ Cl 2 ( π / 3 ) |\operatorname{Cl}_2|\le\operatorname{Cl}_2(\pi/3) ∣ Cl 2 ∣ ≤ Cl 2 ( π /3 ) ± G ( d ⋅ + d / 2 ) \pm G(d\cdot+d/2) ± G ( d ⋅ + d /2 ) [ 0 , X ] [0,X] [ 0 , X ] ≤ κ / ( d X ) \le\kappa/(dX) ≤ κ / ( d X ) G ≤ log 2 G\le\log2 G ≤ log 2 μ = − 1 \mu=-1 μ = − 1 L = d X ≤ 1 L=dX\le1 L = d X ≤ 1 − G -G − G L L L 0 0 0 2 ∣ sin π s ∣ ≥ 4 ∣ s ∣ 2|\sin\pi s|\ge4|s| 2∣ sin π s ∣ ≥ 4∣ s ∣ ≤ log ( e / ( 2 L ) ) \le\log(e/(2L)) ≤ log ( e / ( 2 L )) X = 1 / ( 2 q ) X=1/(2q) X = 1/ ( 2 q ) 2 ω ( q ) − 1 2^{\omega(q)-1} 2 ω ( q ) − 1 μ = + 1 \mu=+1 μ = + 1 ≤ log 2 \le\log2 ≤ log 2 μ = − 1 \mu=-1 μ = − 1 d = q / e d=q/e d = q / e ≤ 1 + log e \le1+\log e ≤ 1 + log e ∑ ω ( e ) odd log e ≤ 2 ω ( q ) − 1 log rad q \sum_{\omega(e)\text{ odd}}\log e\le2^{\omega(q)-1}\log\operatorname{rad}q ∑ ω ( e ) odd log e ≤ 2 ω ( q ) − 1 log rad q κ \kappa κ β \beta β L = 1 L=1 L = 1 β \beta β 1 − log 2 1-\log2 1 − log 2 κ \kappa κ κ > 1 − log 2 \kappa>1-\log2 κ > 1 − log 2 X = 1 / ( 2 q ) X=1/(2q) X = 1/ ( 2 q ) μ = − 1 \mu=-1 μ = − 1 L ≤ 1 / 4 L\le1/4 L ≤ 1/4 β ≥ κ \beta\ge\kappa β ≥ κ R q R_q R q
So ℓ q ≪ τ ( q ) log q / φ ( q ) = q − 1 + o ( 1 ) \ell_q\ll\tau(q)\log q/\varphi(q)=q^{-1+o(1)} ℓ q ≪ τ ( q ) log q / φ ( q ) = q − 1 + o ( 1 )
Corollary Q′ (the largest lobe)
sup q ∉ E ℓ q = ℓ 3 \sup_{q\notin\mathcal{E}}\ell_q=\ell_3 sup q ∈ / E ℓ q = ℓ 3 ℓ 3 ∈ [ 0.131593 , 0.132469 ] \ell_3\in[0.131593,0.132469] ℓ 3 ∈ [ 0.131593 , 0.132469 ] ℓ 3 = 0.1316 \ell_3=0.1316 ℓ 3 = 0.1316
For q > 20000 q>20000 q > 20000 2 ω ( q ) / φ ( q ) ≤ M θ q − θ 2^{\omega(q)}/\varphi(q)\le M_\theta q^{-\theta} 2 ω ( q ) / φ ( q ) ≤ M θ q − θ θ = 0.70 \theta=0.70 θ = 0.70 M 0.70 = 15.14 M_{0.70}=15.14 M 0.70 = 15.14 q q q 10 4 10^4 1 0 4 < ℓ 3 <\ell_3 < ℓ 3
For q ≤ 20000 q\le20000 q ≤ 20000 R q R_q R q < ℓ 3 <\ell_3 < ℓ 3 ≤ 2730 \le2730 ≤ 2730 q = 3 q=3 q = 3
For each of the other 144 moduli, sup X ≥ 1 / ( 2 q ) A q ( X ) / φ ( q ) \sup_{X\ge1/(2q)}A_q(X)/\varphi(q) sup X ≥ 1/ ( 2 q ) A q ( X ) / φ ( q ) < ℓ 3 <\ell_3 < ℓ 3 q = 15 q=15 q = 15 X q ≥ 1 / ( 2 q ) X_q\ge1/(2q) X q ≥ 1/ ( 2 q ) ℓ q \ell_q ℓ q H ( X ) = ∫ 0 X log ∣ Φ q ( − e ( x ) ) ∣ d x = ∑ d ∣ q μ ( q / d ) d − 1 [ C ( d X + d / 2 ) − C ( d / 2 ) ] H(X)=\int_0^X\log|\Phi_q(-e(x))|\,dx=\sum_{d\mid q}\mu(q/d)d^{-1}[C(dX+d/2)-C(d/2)] H ( X ) = ∫ 0 X log ∣ Φ q ( − e ( x )) ∣ d x = ∑ d ∣ q μ ( q / d ) d − 1 [ C ( d X + d /2 ) − C ( d /2 )] C ( u ) = − Cl 2 ( 2 π u ) / ( 2 π ) C(u)=-\operatorname{Cl}_2(2\pi u)/(2\pi) C ( u ) = − Cl 2 ( 2 π u ) / ( 2 π ) H ( X ) ≤ H ( a ) + ( X − a ) ⋅ S H(X)\le H(a)+(X-a)\cdot S H ( X ) ≤ H ( a ) + ( X − a ) ⋅ S S S S sup log ∣ Φ q ( − e ( x ) ) ∣ \sup\log|\Phi_q(-e(x))| sup log ∣ Φ q ( − e ( x )) ∣ ∑ ( r , q ) = 1 log ∣ 2 cos π ( r / q + x ) ∣ \sum_{(r,q)=1}\log|2\cos\pi(r/q+x)| ∑ ( r , q ) = 1 log ∣2 cos π ( r / q + x ) ∣
each term maximised exactly on the interval; beyond X ∗ X^{*} X ∗ ∣ H ( X ) ∣ ≤ κ ∑ d ∣ q , μ ( q / d ) ≠ 0 1 / d |H(X)| \le\kappa\sum_{d \mid q,\mu(q/d)\ne0} 1/d ∣ H ( X ) ∣ ≤ κ ∑ d ∣ q , μ ( q / d ) = 0 1/ d
ℓ 3 \ell_3 ℓ 3 A 3 ( 29 / 40 ) / 2 A_3(29/40)/2 A 3 ( 29/40 ) /2 A 3 A_3 A 3 ( 1 / 5 , 21 / 100 ) (1/5,21/100) ( 1/5 , 21/100 )
The constant M 0.70 M_{0.70} M 0.70 ≤ 15.1373 \le15.1373 ≤ 15.1373 q = 30030 q = 30030 q = 30030 p ≤ 13 p \le13 p ≤ 13 q = 20001 q = 20001 q = 20001 0.0856 0.0856 0.0856
The corollary is proved: the finite part is a certified computation in interval arithmetic (certify_lobes_arb.py, 39 s).
The core of an arc: Gaussian and Edgeworth
Proposition C (core expansion). Put t = 2 π β t = 2\pi\beta t = 2 π β ( 1 + u e i w t ) / ( 1 + u ) (1+ue^{iwt})/(1+u) ( 1 + u e i w t ) / ( 1 + u ) u = e ( w a / q ) u=e(wa/q) u = e ( w a / q ) w ⋅ Bernoulli ( p ) w\cdot\operatorname{Bernoulli}(p) w ⋅ Bernoulli ( p ) p = u / ( 1 + u ) p=u/(1+u) p = u / ( 1 + u ) κ 1 = w p \kappa_1=wp κ 1 = w p κ 2 = w 2 p ( 1 − p ) \kappa_2=w^2p(1-p) κ 2 = w 2 p ( 1 − p ) κ 3 = w 3 p ( 1 − p ) ( 1 − 2 p ) \kappa_3=w^3p(1-p)(1-2p) κ 3 = w 3 p ( 1 − p ) ( 1 − 2 p ) κ 4 = w 4 p ( 1 − p ) ( 1 − 6 p + 6 p 2 ) \kappa_4=w^4p(1-p)(1-6p+6p^2) κ 4 = w 4 p ( 1 − p ) ( 1 − 6 p + 6 p 2 ) ( w = q / 2 ) (w=q/2) ( w = q /2 ) 1 − e i w t = w ( − i t ) e i w t / 2 sinc ( w t / 2 ) 1-e^{iwt}=w(-it)e^{iwt/2}\operatorname{sinc}(wt/2) 1 − e i w t = w ( − i t ) e i w t /2 sinc ( w t /2 ) w ( − i t ) w(-it) w ( − i t ) w ⋅ d / d T w\cdot d/dT w ⋅ d / d T w / 2 w/2 w /2 w 2 / 12 w^2/12 w 2 /12 0 0 0 − w 4 / 120 -w^4/120 − w 4 /120 κ j ( a / q ) \kappa_j(a/q) κ j ( a / q ) s = κ 2 s=\sqrt{\kappa_2} s = κ 2 z = ( T − κ 1 ) / s z=(T-\kappa_1)/s z = ( T − κ 1 ) / s
A a / q ( T ) = e ( − T a / q ) F N ∗ ( a / q ) ( ∏ w ∈ V w d d T ) [ φ ( z ) s ( 1 + κ 3 6 s 3 He 3 ( z ) + κ 4 24 s 4 He 4 ( z ) + κ 3 2 72 s 6 He 6 ( z ) ) ] . \mathcal{A}_{a/q}(T)=e(-Ta/q)F_N^*(a/q)\left(\prod_{w\in\mathcal{V}}w\frac{d}{dT}\right)\left[\frac{\varphi(z)}{s}\left(1+\frac{\kappa_3}{6s^3}\operatorname{He}_3(z)+\frac{\kappa_4}{24s^4}\operatorname{He}_4(z)+\frac{\kappa_3^2}{72s^6}\operatorname{He}_6(z)\right)\right]. A a / q ( T ) = e ( − T a / q ) F N ∗ ( a / q ) ( w ∈ V ∏ w d T d ) [ s φ ( z ) ( 1 + 6 s 3 κ 3 He 3 ( z ) + 24 s 4 κ 4 He 4 ( z ) + 72 s 6 κ 3 2 He 6 ( z ) ) ] . We tested it on the exact counts at N = 500 N=500 N = 500 2 84 2^{84} 2 84 ∑ T r N ( T ) e ( a T / q ) e − ( T − T 0 ) 2 / 2 H 2 \sum_T r_N(T)e(aT/q)e^{-(T-T_0)^2/2H^2} ∑ T r N ( T ) e ( a T / q ) e − ( T − T 0 ) 2 /2 H 2 H = 60 H=60 H = 60 T 0 T_0 T 0
Table 2. Ratio exact/predicted of the arc terms. The phases agree to ≤ 2 ⋅ 10 − 3 \le2\cdot10^{-3} ≤ 2 ⋅ 1 0 − 3
Proposition C′ (the next two orders). Let a non-vanishing factor have, besides κ 1 , … , κ 4 \kappa_1,\ldots,\kappa_4 κ 1 , … , κ 4 κ 5 = w 5 p ( 1 − p ) ( 1 − 2 p ) ( 1 − 12 p + 12 p 2 ) \kappa_5=w^5p(1-p)(1-2p)(1-12p+12p^2) κ 5 = w 5 p ( 1 − p ) ( 1 − 2 p ) ( 1 − 12 p + 12 p 2 ) κ 6 = w 6 p ( 1 − p ) ( 1 − 30 p + 150 p 2 − 240 p 3 + 120 p 4 ) \kappa_6=w^6p(1-p)(1-30p+150p^2-240p^3+120p^4) κ 6 = w 6 p ( 1 − p ) ( 1 − 30 p + 150 p 2 − 240 p 3 + 120 p 4 ) κ 3 = κ 5 = 0 \kappa_3=\kappa_5=0 κ 3 = κ 5 = 0 κ 6 = w 6 / 252 \kappa_6=w^6/252 κ 6 = w 6 /252 s − 3 s^{-3} s − 3
κ 5 120 s 5 H e 5 ( z ) + κ 3 κ 4 144 s 7 H e 7 ( z ) + κ 3 3 1296 s 9 H e 9 ( z ) , \frac{\kappa_{5}}{120s^{5}}He_{5}(z)+\frac{\kappa_{3}\kappa_{4}}{144s^{7}}He_{7}(z)+\frac{\kappa_{3}^{3}}{1296s^{9}}He_{9}(z), 120 s 5 κ 5 H e 5 ( z ) + 144 s 7 κ 3 κ 4 H e 7 ( z ) + 1296 s 9 κ 3 3 H e 9 ( z ) , and of order s − 4 s^{-4} s − 4
κ 6 720 s 6 H e 6 ( z ) + ( κ 4 2 1152 + κ 3 κ 5 720 ) H e 8 ( z ) s 8 + κ 3 2 κ 4 1728 s 10 H e 10 ( z ) + κ 3 4 31104 s 12 H e 12 ( z ) . \frac{\kappa_{6}}{720s^{6}}He_{6}(z)+\left(\frac{\kappa_{4}^{2}}{1152}+\frac{\kappa_{3}\kappa_{5}}{720}\right)\frac{He_{8}(z)}{s^{8}}+\frac{\kappa_{3}^{2}\kappa_{4}}{1728s^{10}}He_{10}(z)+\frac{\kappa_{3}^{4}}{31104s^{12}}He_{12}(z). 720 s 6 κ 6 H e 6 ( z ) + ( 1152 κ 4 2 + 720 κ 3 κ 5 ) s 8 H e 8 ( z ) + 1728 s 10 κ 3 2 κ 4 H e 10 ( z ) + 31104 s 12 κ 3 4 H e 12 ( z ) . Proof. These are the terms of order s − 3 s^{-3} s − 3 s − 4 s^{-4} s − 4 w ⋅ Bernoulli ( p ) w\cdot\operatorname{Bernoulli}(p) w ⋅ Bernoulli ( p ) log ( 1 − p + p e i w t ) \log(1-p+pe^{iwt}) log ( 1 − p + p e i w t ) e i w t / 2 sinc ( w t / 2 ) e^{iwt/2}\operatorname{sinc}(wt/2) e i w t /2 sinc ( w t /2 ) [ 0 , w ] [0,w] [ 0 , w ] w 6 B 6 / 6 = w 6 / 252 w^{6}B_{6}/6=w^{6}/252 w 6 B 6 /6 = w 6 /252
Table 2′. The same test with H = 120 H=120 H = 120 H = 60 H=60 H = 60 T 0 T_{0} T 0 H ≥ 120 H\ge120 H ≥ 120
The arc expansion as a theorem
Fix B > 40 B>40 B > 40 P = ( log N ) B / 4 \mathfrak{P}=(\log N)^{B/4} P = ( log N ) B /4 M = ( log N ) B / 4 M=(\log N)^{B/4} M = ( log N ) B /4 M ( a / q ) = { θ : ∣ θ − a / q ∣ ≤ P / ( q N ) } \mathfrak{M}(a/q)=\{\theta:|\theta-a/q|\le\mathfrak{P}/(qN)\} M ( a / q ) = { θ : ∣ θ − a / q ∣ ≤ P / ( q N )} q ≤ P q\le\mathfrak{P} q ≤ P m \mathfrak{m} m q ≥ 3 q\ge3 q ≥ 3 q = 1 q=1 q = 1 q = 2 q=2 q = 2 R a / q ( T ) = ∫ M ( a / q ) F N ( e ( θ ) ) e ( − T θ ) d θ R_{a/q}(T)=\int_{\mathfrak{M}(a/q)}F_N(e(\theta))e(-T\theta)\,d\theta R a / q ( T ) = ∫ M ( a / q ) F N ( e ( θ )) e ( − T θ ) d θ r N ( T ) = ∑ q ≤ P ∑ a R a / q ( T ) + ∫ m F N ( e ( θ ) ) e ( − T θ ) d θ r_N(T)=\sum_{q\le\mathfrak{P}}\sum_aR_{a/q}(T)+\int_{\mathfrak{m}}F_N(e(\theta))e(-T\theta)\,d\theta r N ( T ) = ∑ q ≤ P ∑ a R a / q ( T ) + ∫ m F N ( e ( θ )) e ( − T θ ) d θ
Lemma M (one-sided trigonometric majorant). For every M ≥ 2 M\ge2 M ≥ 2 K M K_M K M ≤ M \le M ≤ M K M ( x ) ≥ log ∣ cos π x ∣ K_M(x)\ge\log|\cos\pi x| K M ( x ) ≥ log ∣ cos π x ∣ x x x K ^ M ( 0 ) ≤ − log 2 + C M − 1 / 2 \widehat{K}_M(0)\le-\log2+CM^{-1/2} K M ( 0 ) ≤ − log 2 + C M − 1/2 ∣ K ^ M ( m ) ∣ ≤ C |\widehat{K}_M(m)|\le C ∣ K M ( m ) ∣ ≤ C
Proof. Let f = log ∣ cos π x ∣ f=\log|\cos\pi x| f = log ∣ cos π x ∣ f ε = max ( f , log ε ) f_{\varepsilon}=\max(f,\log\varepsilon) f ε = max ( f , log ε ) f ≤ f ε f\le f_{\varepsilon} f ≤ f ε f ^ ε ( 0 ) = − log 2 + O ( ε ) \widehat{f}_{\varepsilon}(0)=-\log2+O(\varepsilon) f ε ( 0 ) = − log 2 + O ( ε ) f ε f_{\varepsilon} f ε π / ε \pi/\varepsilon π / ε J M J_M J M K M = J M ∗ f ε + C π / ( ε M ) K_M=J_M*f_{\varepsilon}+C\pi/(\varepsilon M) K M = J M ∗ f ε + C π / ( εM ) f ε f_{\varepsilon} f ε ε = M − 1 / 2 \varepsilon=M^{-1/2} ε = M − 1/2 J M ≥ 0 J_M\ge0 J M ≥ 0 ∣ K ^ M ( m ) ∣ ≤ ∫ 0 1 ∣ f ε ∣ ≤ log 2 + O ( ε ) |\widehat{K}_M(m)|\le\int_{0}^{1}|f_{\varepsilon}|\le\log2+O(\varepsilon) ∣ K M ( m ) ∣ ≤ ∫ 0 1 ∣ f ε ∣ ≤ log 2 + O ( ε ) m ≠ 0 m\ne0 m = 0
An L 1 L^{1} L 1 f f f − ∞ -\infty − ∞
Lemma L (neighbourhood of an arc). Fix B B B 3 ≤ q ≤ P 3\le q\le\mathfrak{P} 3 ≤ q ≤ P ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 θ = a / q + X / N \theta=a/q+X/N θ = a / q + X / N N − 1 / 2 ≤ ∣ X ∣ ≤ P / q N^{-1/2}\le|X|\le\mathfrak{P}/q N − 1/2 ≤ ∣ X ∣ ≤ P / q
log ∣ F N ( e ( θ ) ) ∣ ≤ n log 2 + N log N ( A q ( ∣ X ∣ ) φ ( q ) − log 2 ) + O B ( N log 2 N ) . \log|F_N(e(\theta))| \le n\log2+\frac{N}{\log N}\left(\frac{A_q(|X|)}{\varphi(q)}-\log2\right)+O_B\left(\frac{N}{\log^2 N}\right). log ∣ F N ( e ( θ )) ∣ ≤ n log 2 + log N N ( φ ( q ) A q ( ∣ X ∣ ) − log 2 ) + O B ( log 2 N N ) . Only this upper bound is used below.
Proof. Write log ∣ 1 + e ( w θ ) ∣ = log 2 + g ( w θ ) \log|1+e(w\theta)|=\log2+g(w\theta) log ∣1 + e ( w θ ) ∣ = log 2 + g ( w θ ) g ( y ) = log ∣ cos π y ∣ ≤ 0 g(y)=\log|\cos\pi y|\le0 g ( y ) = log ∣ cos π y ∣ ≤ 0 ∣ Φ q ( − e ( − x ) ) ∣ = ∣ Φ q ( − e ( x ) ) ∣ |\Phi_q(-e(-x))|=|\Phi_q(-e(x))| ∣ Φ q ( − e ( − x )) ∣ = ∣ Φ q ( − e ( x )) ∣ X > 0 X>0 X > 0 β = X / N \beta=X/N β = X / N
If p ≡ r ( m o d q ) p\equiv r\pmod q p ≡ r ( mod q ) p a / q − r a / q pa/q-ra/q p a / q − r a / q ∣ cos π p θ ∣ = ∣ cos π ( r a / q + p β ) ∣ |\cos\pi p\theta|=|\cos\pi(ra/q+p\beta)| ∣ cos π pθ ∣ = ∣ cos π ( r a / q + pβ ) ∣ g r ( x ) = g ( r a / q + x ) g_r(x)=g(ra/q+x) g r ( x ) = g ( r a / q + x ) g r , ε = max ( g r , log ε ) g_{r,\varepsilon}=\max(g_r,\log\varepsilon) g r , ε = max ( g r , log ε ) ε = 1 / N \varepsilon=1/N ε = 1/ N g r ≤ g r , ε ≤ 0 g_r\le g_{r,\varepsilon}\le0 g r ≤ g r , ε ≤ 0 [ 0 , X ] [0,X] [ 0 , X ] g r , ε g_{r,\varepsilon} g r , ε 2 X + 4 2X+4 2 X + 4 ≤ log N \le\log N ≤ log N
By Siegel–Walfisz, π ( t ; q , r ) = li ( t ) / φ ( q ) + E r ( t ) \pi(t;q,r)=\operatorname{li}(t)/\varphi(q)+E_r(t) π ( t ; q , r ) = li ( t ) / φ ( q ) + E r ( t ) ∣ E r ( t ) ∣ ≪ B N e − c B log N |E_r(t)|\ll_B Ne^{-c_B\sqrt{\log N}} ∣ E r ( t ) ∣ ≪ B N e − c B l o g N N ≤ t ≤ N \sqrt{N}\le t\le N N ≤ t ≤ N q ≤ P q\le\mathfrak{P} q ≤ P 2 B 2B 2 B P ≤ ( log t ) 2 B \mathfrak{P}\le(\log t)^{2B} P ≤ ( log t ) 2 B t ≥ N t\ge\sqrt{N} t ≥ N N N N ∣ ∫ f d E ∣ ≤ ( 2 sup ∣ f ∣ + Var f ) ⋅ sup ∣ E ∣ |\int f\,dE|\le(2\sup|f|+\operatorname{Var}f)\cdot\sup|E| ∣ ∫ f d E ∣ ≤ ( 2 sup ∣ f ∣ + Var f ) ⋅ sup ∣ E ∣
∑ N < p ≤ N p ≡ r ( m o d q ) g r , ε ( p β ) = 1 φ ( q ) ∫ N N g r , ε ( t β ) d t log t + O B ( ( X + 3 ) N log N e − c B log N ) . \sum_{\substack{\sqrt{N}<p\le N\\p\equiv r\pmod q}}g_{r,\varepsilon}(p\beta)
=\frac{1}{\varphi(q)}\int_{\sqrt{N}}^N g_{r,\varepsilon}(t\beta)\frac{dt}{\log t}
+O_B\left((X+3)N\log N e^{-c_B\sqrt{\log N}}\right). N < p ≤ N p ≡ r ( mod q ) ∑ g r , ε ( pβ ) = φ ( q ) 1 ∫ N N g r , ε ( tβ ) log t d t + O B ( ( X + 3 ) N log N e − c B l o g N ) . Since g r , ε ≤ 0 g_{r,\varepsilon}\le0 g r , ε ≤ 0 log t ≤ log N \log t\le\log N log t ≤ log N
1 log N ∫ N N g r , ε ( t β ) d t = N X log N ∫ X / N X g r , ε ( x ) d x ≤ N X log N [ ∫ 0 X g r + ∫ 0 X ( g r , ε − g r ) + X log N N ] . \frac{1}{\log N}\int_{\sqrt{N}}^N g_{r,\varepsilon}(t\beta)\,dt
=\frac{N}{X\log N}\int_{X/\sqrt{N}}^X g_{r,\varepsilon}(x)\,dx
\le\frac{N}{X\log N}\left[\int_0^X g_r+\int_0^X(g_{r,\varepsilon}-g_r)+\frac{X\log N}{\sqrt{N}}\right]. log N 1 ∫ N N g r , ε ( tβ ) d t = X log N N ∫ X / N X g r , ε ( x ) d x ≤ X log N N [ ∫ 0 X g r + ∫ 0 X ( g r , ε − g r ) + N X log N ] . On each period, g r , ε − g r g_{r,\varepsilon}-g_r g r , ε − g r ∣ cos ∣ < ε |\cos|<\varepsilon ∣ cos ∣ < ε 2 ε 2\varepsilon 2 ε ∫ 0 X ( g r , ε − g r ) ≤ 2 ( X + 1 ) / N \int_0^X(g_{r,\varepsilon}-g_r)\le2(X+1)/N ∫ 0 X ( g r , ε − g r ) ≤ 2 ( X + 1 ) / N
By ∏ ( r , q ) = 1 ( 1 + ζ q r a y ) = Φ q ( − y ) \prod_{(r,q)=1}(1+\zeta_q^{ra}y)=\Phi_q(-y) ∏ ( r , q ) = 1 ( 1 + ζ q r a y ) = Φ q ( − y ) ∣ 1 + e ( y ) ∣ = 2 ∣ cos π y ∣ |1+e(y)|=2|\cos\pi y| ∣1 + e ( y ) ∣ = 2∣ cos π y ∣ ∑ ( r , q ) = 1 g r ( x ) = log ∣ Φ q ( − e ( x ) ) ∣ − φ ( q ) log 2 \sum_{(r,q)=1}g_r(x)=\log|\Phi_q(-e(x))|-\varphi(q)\log2 ∑ ( r , q ) = 1 g r ( x ) = log ∣ Φ q ( − e ( x )) ∣ − φ ( q ) log 2 ∑ r ∫ 0 X g r = X ( A q ( X ) − φ ( q ) log 2 ) \sum_r\int_0^X g_r=X(A_q(X)-\varphi(q)\log2) ∑ r ∫ 0 X g r = X ( A q ( X ) − φ ( q ) log 2 )
∑ w g ( w θ ) ≤ N log N ( A q ( X ) φ ( q ) − log 2 ) + O ( N log N ) + O ( N ) + O B ( P 2 N log N e − c B log N ) , \sum_w g(w\theta)\le\frac{N}{\log N}\left(\frac{A_q(X)}{\varphi(q)}-\log2\right)+O\left(\frac{\sqrt{N}}{\log N}\right)+O(\sqrt{N})+O_B\left(\mathfrak{P}^2N\log N e^{-c_B\sqrt{\log N}}\right), w ∑ g ( w θ ) ≤ log N N ( φ ( q ) A q ( X ) − log 2 ) + O ( log N N ) + O ( N ) + O B ( P 2 N log N e − c B l o g N ) , using X ≥ N − 1 / 2 X\ge N^{-1/2} X ≥ N − 1/2 X ≤ P / q X\le\mathfrak{P}/q X ≤ P / q φ ( q ) ≤ P \varphi(q)\le\mathfrak{P} φ ( q ) ≤ P O B ( N / log 2 N ) O_B(N/\log^2N) O B ( N / log 2 N )
Lemma L0 _0 0 For 3 ≤ q ≤ P 3\le q\le\mathfrak{P} 3 ≤ q ≤ P ( a , q ) = 1 (a,q)=1 ( a , q ) = 1 θ = a / q + β \theta=a/q+\beta θ = a / q + β ∣ β ∣ ≤ N − 3 / 2 |\beta|\le N^{-3/2} ∣ β ∣ ≤ N − 3/2
log ∣ F N ( e ( θ ) ) ∣ ≤ log 2 + log ∣ F N ∗ ( a / q ) ∣ + π q π ( N ) N − 1 / 2 . \log|F_N(e(\theta))|\le\log2+\log|F_N^*(a/q)|+\pi q\pi(N)N^{-1/2}. log ∣ F N ( e ( θ )) ∣ ≤ log 2 + log ∣ F N ∗ ( a / q ) ∣ + π q π ( N ) N − 1/2 . Proof. For a weight w ∉ V ( a / q ) w \notin\mathcal{V}(a/q) w ∈ / V ( a / q ) ∥ w a / q − 1 / 2 ∥ ≥ 1 / ( 2 q ) \lVert wa/q-1/2\rVert\ge1/(2q) ∥ w a / q − 1/2 ∥ ≥ 1/ ( 2 q ) ∣ cos ( π w a / q ) ∣ ≥ sin ( π / ( 2 q ) ) ≥ 1 / q \lvert\cos(\pi wa/q)\rvert\ge\sin(\pi/(2q)) \ge1/q ∣ cos ( π w a / q )∣ ≥ sin ( π / ( 2 q )) ≥ 1/ q ∣ cos π ( w a / q + w β ) ∣ ≤ ∣ cos ( π w a / q ) ∣ + π w ∣ β ∣ ≤ ∣ cos ( π w a / q ) ∣ ( 1 + π q w ∣ β ∣ ) \lvert\cos\pi(wa/q+w\beta)\rvert\le\lvert\cos(\pi wa/q)\rvert+\pi w\lvert\beta\rvert\le\lvert\cos(\pi wa/q)\rvert(1+\pi qw\lvert\beta\rvert) ∣ cos π ( w a / q + w β )∣ ≤ ∣ cos ( π w a / q )∣ + π w ∣ β ∣ ≤ ∣ cos ( π w a / q )∣ ( 1 + π q w ∣ β ∣) ∑ w π q w ∣ β ∣ ≤ π q π ( N ) N ∣ β ∣ ≤ π q π ( N ) N − 1 / 2 \sum_w\pi qw\lvert\beta\rvert\le\pi q\pi(N)N\lvert\beta\rvert\le\pi q\pi(N)N^{-1/2} ∑ w π q w ∣ β ∣ ≤ π q π ( N ) N ∣ β ∣ ≤ π q π ( N ) N − 1/2 ≤ 2 \le2 ≤ 2
The peak of an arc
For q ≥ 3 q \ge3 q ≥ 3 G ( θ ) G(\theta) G ( θ ) F N ( e ( θ ) ) F_N(e(\theta)) F N ( e ( θ )) G ( a / q ) = F N ∗ ( a / q ) G(a/q)=F_N^*(a/q) G ( a / q ) = F N ∗ ( a / q ) t = 2 π ( θ − a / q ) t=2\pi(\theta-a/q) t = 2 π ( θ − a / q ) S 2 = ∑ w ∉ V w 2 S_2=\sum_{w\notin\mathcal{V}}w^2 S 2 = ∑ w ∈ / V w 2
ν a / q = 1 2 ∑ w ∉ V w tan π w a q . \nu_{a/q}=\frac{1}{2}\sum_{w\notin\mathcal{V}}w\tan\frac{\pi wa}{q}. ν a / q = 2 1 w ∈ / V ∑ w tan q π w a . Lemma W (the peak). Fix B B B 3 ≤ q ≤ P 3\le q\le P 3 ≤ q ≤ P ( a , q ) = 1 (a,q)=1 ( a , q ) = 1
On ∣ t ∣ < π / ( q N ) \lvert t\rvert<\pi/(qN) ∣ t ∣ < π / ( q N ) G G G t ↦ log ∣ G ∣ t\mapsto\log\lvert G\rvert t ↦ log ∣ G ∣ ≤ − S 2 / 4 \le-S_2/4 ≤ − S 2 /4
Its derivative at t = 0 t=0 t = 0 − ν a / q -\nu_{a/q} − ν a / q ν a / q ≪ B N 2 e − c B log N \nu_{a/q}\ll_B N^2e^{-c_B\sqrt{\log N}} ν a / q ≪ B N 2 e − c B l o g N S 2 ≥ N 3 / ( 64 log N ) S_2\ge N^3/(64\log N) S 2 ≥ N 3 / ( 64 log N ) N N N ( N / 2 , N ] (N/2,N] ( N /2 , N ] q = 1 q=1 q = 1 log ∣ G ∣ \log\lvert G\rvert log ∣ G ∣ t ∗ t^* t ∗ ∣ t ∗ ∣ ≤ 4 ∣ ν a / q ∣ / S 2 ≪ B N − 1 e − c B log N \lvert t^*\rvert\le4\lvert\nu_{a/q}\rvert/S_2\ll_B N^{-1}e^{-c_B\sqrt{\log N}} ∣ t ∗ ∣ ≤ 4 ∣ ν a / q ∣ / S 2 ≪ B N − 1 e − c B l o g N
At θ ∗ = a / q + t ∗ / ( 2 π ) \theta^*=a/q+t^*/(2\pi) θ ∗ = a / q + t ∗ / ( 2 π ) u = e ( w θ ∗ ) u=e(w\theta^*) u = e ( w θ ∗ ) κ 1 ∗ \kappa_1^* κ 1 ∗ κ 2 ∗ = ∑ w ∉ V w 2 / ( 4 cos 2 ( π w θ ∗ ) ) ≥ S 2 / 4 \kappa_2^*=\sum_{w\notin\mathcal{V}}w^2/(4\cos^2(\pi w\theta^*))\ge S_2/4 κ 2 ∗ = ∑ w ∈ / V w 2 / ( 4 cos 2 ( π w θ ∗ )) ≥ S 2 /4 κ 3 ∗ \kappa_3^* κ 3 ∗ κ 4 ∗ \kappa_4^* κ 4 ∗ (a/q)\rvert\le\log\lvert G(\theta^ )\rvert\le\log\lvert F_N^*(a/q)\rvert+4\nu_{a/q}^2/S_2.
Proof.
(1) The factor w w w w θ ≡ 1 / 2 ( m o d 1 ) w\theta\equiv1/2\pmod1 w θ ≡ 1/2 ( mod 1 ) w ∉ V w\notin\mathcal{V} w ∈ / V ∥ w a / q − 1 / 2 ∥ ≥ 1 / ( 2 q ) \lVert wa/q-1/2\rVert\ge1/(2q) ∥ w a / q − 1/2 ∥ ≥ 1/ ( 2 q ) ≥ 1 / ( 2 q w ) ≥ 1 / ( 2 q N ) \ge1/(2qw)\ge1/(2qN) ≥ 1/ ( 2 q w ) ≥ 1/ ( 2 q N ) a / q a/q a / q d 2 / d t 2 log ∣ cos ( π w θ ) ∣ = − w 2 / ( 4 cos 2 ( π w θ ) ) ≤ − w 2 / 4 d^2/dt^2\log\lvert\cos(\pi w\theta)\rvert=-w^2/(4\cos^2(\pi w\theta))\le-w^2/4 d 2 / d t 2 log ∣ cos ( π w θ )∣ = − w 2 / ( 4 cos 2 ( π w θ )) ≤ − w 2 /4
(2) With u = e ( w a / q ) u=e(wa/q) u = e ( w a / q ) u / ( 1 + u ) = 1 / 2 + ( i / 2 ) tan ( π w a / q ) u/(1+u)=1/2+(i/2)\tan(\pi wa/q) u / ( 1 + u ) = 1/2 + ( i /2 ) tan ( π w a / q ) d / d t log ∣ 1 + u e i w t ∣ d/dt\log\lvert1+ue^{iwt}\rvert d / d t log ∣ 1 + u e i w t ∣ t = 0 t=0 t = 0 − w ⋅ Im ( u / ( 1 + u ) ) = − ( w / 2 ) tan ( π w a / q ) -w\cdot\operatorname{Im}(u/(1+u))=-(w/2)\tan(\pi wa/q) − w ⋅ Im ( u / ( 1 + u )) = − ( w /2 ) tan ( π w a / q ) ν \nu ν ∑ p ≡ r , p ≤ N p = φ ( q ) − 1 ∑ p ≤ N , p ∤ q p + O B ( N 2 e − c B log N ) \sum_{p\equiv r,p\le N}p=\varphi(q)^{-1}\sum_{p\le N,p\nmid q}p+O_B(N^2e^{-c_B\sqrt{\log N}}) ∑ p ≡ r , p ≤ N p = φ ( q ) − 1 ∑ p ≤ N , p ∤ q p + O B ( N 2 e − c B l o g N ) ( log N ) 2 B (\log N)^{2B} ( log N ) 2 B t ≥ N t\ge\sqrt N t ≥ N p ≤ N p\le\sqrt N p ≤ N O ( N ) O(N) O ( N ) ∑ ( r , q ) = 1 tan ( π r a / q ) = 0 \sum_{(r,q)=1}\tan(\pi ra/q)=0 ∑ ( r , q ) = 1 tan ( π r a / q ) = 0 r r r q − r q-r q − r ∣ tan ( π r a / q ) ∣ ≤ q \lvert\tan(\pi ra/q)\rvert\le q ∣ tan ( π r a / q )∣ ≤ q 1 1 1 q q q O ( q 2 log q ) O(q^2\log q) O ( q 2 log q ) 4 ∣ ν ∣ / S 2 4\lvert\nu\rvert/S_2 4 ∣ ν ∣ / S 2 0 0 0 N N N
(3) On ∣ u ∣ = 1 \lvert u\rvert=1 ∣ u ∣ = 1 p ( 1 − p ) = u / ( 1 + u ) 2 = 1 / ( 4 cos 2 ( π w θ ) ) p(1-p)=u/(1+u)^2=1/(4\cos^2(\pi w\theta)) p ( 1 − p ) = u / ( 1 + u ) 2 = 1/ ( 4 cos 2 ( π w θ )) 1 − 2 p = − i tan ( π w θ ) 1-2p=-i\tan(\pi w\theta) 1 − 2 p = − i tan ( π w θ ) κ 2 ∗ , κ 3 ∗ , κ 4 ∗ \kappa_2^*,\kappa_3^*,\kappa_4^* κ 2 ∗ , κ 3 ∗ , κ 4 ∗ κ 1 ∗ \kappa_1^* κ 1 ∗ − Im κ 1 ∗ -\operatorname{Im}\kappa_1^* − Im κ 1 ∗ log ∣ G ∣ \log\lvert G\rvert log ∣ G ∣ t ∗ t^* t ∗ log ∣ G ( θ ∗ ) ∣ − log ∣ G ( a / q ) ∣ ≤ ∣ ν ∣ ∣ t ∗ ∣ ≤ 4 ν 2 / S 2 \log\lvert G(\theta^*)\rvert-\log\lvert G(a/q)\rvert\le\lvert\nu\rvert\lvert t^*\rvert\le4\nu^2/S_2 log ∣ G ( θ ∗ )∣ − log ∣ G ( a / q )∣ ≤ ∣ ν ∣ ∣ t ∗ ∣ ≤ 4 ν 2 / S 2
The offset t ∗ t^* t ∗ ν a / q \nu_{a/q} ν a / q r r r q q q tan ( π r a / q ) \tan(\pi ra/q) tan ( π r a / q ) q q q q q q ν a / q ≪ q log q ⋅ N 3 / 2 log 2 ( q N ) \nu_{a/q}\ll q\log q\cdot N^{3/2}\log^2(qN) ν a / q ≪ q log q ⋅ N 3/2 log 2 ( q N ) ∣ t ∗ ∣ ≪ q log q ⋅ N − 3 / 2 log 3 ( q N ) \lvert t^*\rvert\ll q\log q\cdot N^{-3/2}\log^3(qN) ∣ t ∗ ∣ ≪ q log q ⋅ N − 3/2 log 3 ( q N )
Theorem E (arc expansion). Fix B > 40 B>40 B > 40 B = 48 B=48 B = 48 O ( 1 / log N ) O(1/\log N) O ( 1/ log N ) 0 < ε < 1 / 8 0<\varepsilon<1/8 0 < ε < 1/8 ε \varepsilon ε N → ∞ N\to\infty N → ∞ 3 ≤ q ≤ P 3\le q\le\mathcal{P} 3 ≤ q ≤ P
(minor arcs) sup θ ∈ m ∣ F N ( e ( θ ) ) ∣ ≤ 2 n exp ( ( − log 2 + O ( ( log N ) 5 − B / 8 ) + O ( M − 1 / 2 ) ) π ( N ) ) \sup_{\theta\in\mathfrak{m}}|F_N(e(\theta))|\le2^n\exp((-\log2+O((\log N)^{5-B/8})+O(M^{-1/2}))\pi(N)) sup θ ∈ m ∣ F N ( e ( θ )) ∣ ≤ 2 n exp (( − log 2 + O (( log N ) 5 − B /8 ) + O ( M − 1/2 )) π ( N ))
(emerging arcs) Let q ∈ E q\in\mathcal{E} q ∈ E κ j ∗ \kappa_j^* κ j ∗ s ∗ = κ 2 ∗ s^*=\sqrt{\kappa_2^*} s ∗ = κ 2 ∗ z = ( T − κ 1 ∗ ) / s ∗ z=(T-\kappa_1^*)/s^* z = ( T − κ 1 ∗ ) / s ∗
H ^ ( T ) = e − i T t ∗ G ( θ ∗ ) φ ( z ) s ∗ ( 1 + κ 3 ∗ 6 s ∗ 3 H e 3 ( z ) + κ 4 ∗ 24 s ∗ 4 H e 4 ( z ) + κ 3 ∗ 2 72 s ∗ 6 H e 6 ( z ) ) . \widehat{H}(T)=e^{-iTt^*}G(\theta^*)\frac{\varphi(z)}{s^*}\left(1+\frac{\kappa_3^*}{6s^{*3}}He_3(z)+\frac{\kappa_4^*}{24s^{*4}}He_4(z)+\frac{\kappa_3^{*2}}{72s^{*6}}He_6(z)\right). H ( T ) = e − i T t ∗ G ( θ ∗ ) s ∗ φ ( z ) ( 1 + 6 s ∗ 3 κ 3 ∗ H e 3 ( z ) + 24 s ∗ 4 κ 4 ∗ H e 4 ( z ) + 72 s ∗ 6 κ 3 ∗ 2 H e 6 ( z ) ) . If V ( a / q ) = ∅ \mathcal{V}(a/q)=\varnothing V ( a / q ) = ∅ A a / q ∗ ( T ) = e ( − T a / q ) H ^ ( T ) \mathcal{A}_{a/q}^*(T)=e(-Ta/q)\widehat{H}(T) A a / q ∗ ( T ) = e ( − T a / q ) H ( T ) Ω = 1 \Omega=1 Ω = 1 V ( a / q ) = { w } \mathcal{V}(a/q)=\{w\} V ( a / q ) = { w } 1 + e ( w θ ) = − e − i w t ⋅ e i w t / 2 sinc ( w t / 2 ) 1+e(w\theta)=-e^{-iwt}\cdot e^{iwt/2}\operatorname{sinc}(wt/2) 1 + e ( w θ ) = − e − i w t ⋅ e i w t /2 sinc ( w t /2 ) G G G G ~ = G ⋅ e i w t / 2 sinc ( w t / 2 ) \widetilde{G}=G\cdot e^{iwt/2}\operatorname{sinc}(wt/2) G = G ⋅ e i w t /2 sinc ( w t /2 ) H ^ \widehat{H} H G ~ ( θ ∗ ) = G ( θ ∗ ) e i w t ∗ / 2 sinc ( w t ∗ / 2 ) \widetilde{G}(\theta^*)=G(\theta^*)e^{iwt^*/2}\operatorname{sinc}(wt^*/2) G ( θ ∗ ) = G ( θ ∗ ) e i w t ∗ /2 sinc ( w t ∗ /2 ) G ( θ ∗ ) G(\theta^*) G ( θ ∗ ) G ~ \widetilde{G} G t ∗ t^* t ∗ G G G e i w t / 2 sinc ( w t / 2 ) e^{iwt/2}\operatorname{sinc}(wt/2) e i w t /2 sinc ( w t /2 ) t ∗ t^* t ∗ κ j ∗ \kappa_j^* κ j ∗ A a / q ∗ ( T ) = e ( − T a / q ) ⋅ w ⋅ ( d / d T ) H ^ ( T ) \mathcal{A}_{a/q}^*(T)=e(-Ta/q)\cdot w\cdot(d/dT)\widehat{H}(T) A a / q ∗ ( T ) = e ( − T a / q ) ⋅ w ⋅ ( d / d T ) H ( T ) Ω = w ( ∣ t ∗ ∣ + 1 / s ~ ∗ ) \Omega=w(|t^*|+1/\widetilde{s}^*) Ω = w ( ∣ t ∗ ∣ + 1/ s ∗ ) s ~ ∗ \widetilde{s}^* s ∗ G ~ \widetilde{G} G t ∗ t^* t ∗ s ~ ∗ \widetilde{s}^* s ∗ s ∗ s^* s ∗ H ^ \widehat{H} H Ω \Omega Ω θ ∗ \theta^* θ ∗ ∣ G ∣ |G| ∣ G ∣ T T T
R a / q ( T ) = A a / q ∗ ( T ) + O ( Ω ∣ G ( θ ∗ ) ∣ N − 3 / 2 + 2 ε s ∗ ) + O ( 2 n e ( − log 2 + ℓ q ∗ + o ( 1 ) ) π ( N ) ) , ℓ q ∗ = sup X ≥ 1 / ( 2 q ) A q ( X ) φ ( q ) . R_{a/q}(T)=\mathcal{A}_{a/q}^*(T)+O\left(\Omega|G(\theta^*)|\frac{N^{-3/2+2\varepsilon}}{s^*}\right)+O\left(2^n e^{(-\log2+\ell_q^*+o(1))\pi(N)}\right),\qquad\ell_q^*=\sup_{X\ge1/(2q)}\frac{A_q(X)}{\varphi(q)}. R a / q ( T ) = A a / q ∗ ( T ) + O ( Ω∣ G ( θ ∗ ) ∣ s ∗ N − 3/2 + 2 ε ) + O ( 2 n e ( − l o g 2 + ℓ q ∗ + o ( 1 )) π ( N ) ) , ℓ q ∗ = X ≥ 1/ ( 2 q ) sup φ ( q ) A q ( X ) . In particular, if V ( a / q ) = ∅ \mathcal{V}(a/q)=\varnothing V ( a / q ) = ∅ R a / q ( T ) = A a / q ∗ ( T ) ( 1 + O ( N − 1 + 2 ε ) ) + O ( 2 n e ( − log 2 + ℓ q ∗ + o ( 1 ) ) π ( N ) ) R_{a/q}(T)=\mathcal{A}_{a/q}^*(T)(1+O(N^{-1+2\varepsilon}))+O(2^n e^{(-\log2+\ell_q^*+o(1))\pi(N)}) R a / q ( T ) = A a / q ∗ ( T ) ( 1 + O ( N − 1 + 2 ε )) + O ( 2 n e ( − l o g 2 + ℓ q ∗ + o ( 1 )) π ( N ) ) ∣ z ∣ ≤ log N |z|\le\sqrt{\log N} ∣ z ∣ ≤ log N
(other arcs) For q ∉ E q\notin\mathcal{E} q ∈ / E ∣ R a / q ( T ) ∣ ≤ 2 n exp ( ( − log 2 + ℓ q ∗ + o ( 1 ) ) π ( N ) ) |R_{a/q}(T)|\le2^n\exp((-\log2+\ell_q^*+o(1))\pi(N)) ∣ R a / q ( T ) ∣ ≤ 2 n exp (( − log 2 + ℓ q ∗ + o ( 1 )) π ( N )) ℓ q ∗ = ℓ q \ell_q^*=\ell_q ℓ q ∗ = ℓ q A q A_q A q X q X_q X q
Note (the cumulants of the sinc factor). At t ∗ t^* t ∗ e i w t / 2 sinc ( w t / 2 ) e^{iwt/2}\operatorname{sinc}(wt/2) e i w t /2 sinc ( w t /2 ) w / 2 + i w 2 t ∗ / 12 w/2+iw^2t^*/12 w /2 + i w 2 t ∗ /12 w 2 / 12 + w 4 t ∗ 2 / 240 w^2/12+w^4t^{*2}/240 w 2 /12 + w 4 t ∗ 2 /240 − i w 4 t ∗ / 120 -iw^4t^*/120 − i w 4 t ∗ /120 − w 4 / 120 − w 6 t ∗ 2 / 504 -w^4/120-w^6t^{*2}/504 − w 4 /120 − w 6 t ∗ 2 /504 w = q / 2 ≤ P w=q/2\le\mathcal{P} w = q /2 ≤ P ∣ t ∗ ∣ ≪ N − 1 e − c log N |t^*|\ll N^{-1}e^{-c\sqrt{\log N}} ∣ t ∗ ∣ ≪ N − 1 e − c l o g N w 2 ∣ t ∗ ∣ ≤ 1 w^2|t^*|\le1 w 2 ∣ t ∗ ∣ ≤ 1 N N N w / 2 w/2 w /2 w 2 / 12 w^2/12 w 2 /12 0 0 0 − w 4 / 120 -w^4/120 − w 4 /120 O ( w 2 ∣ t ∗ ∣ ( 1 + w 2 ) ) O(w^2|t^*|(1+w^2)) O ( w 2 ∣ t ∗ ∣ ( 1 + w 2 ))
Remark (the lobe exponent). Suppose A q A_q A q X q X_q X q 1 / ( 2 q ) ≤ X q 1/(2q)\le X_q 1/ ( 2 q ) ≤ X q A q A_q A q ( 0 , X q ] (0,X_q] ( 0 , X q ] A q A_q A q ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) ( 0 , 1 / ( 2 q ) ) (0,1/(2q)) ( 0 , 1/ ( 2 q )) A q ( y ) > A q ( x ) A_q(y)>A_q(x) A q ( y ) > A q ( x ) 0 < x < y ≤ X q 0<x<y\le X_q 0 < x < y ≤ X q A q A_q A q [ x ′ , y ] [x',y] [ x ′ , y ] x ′ < x x'<x x ′ < x X q X_q X q ℓ q ∗ = max ( A q ( 1 / ( 2 q ) ) / φ ( q ) , ℓ q ) \ell_q^*=\max(A_q(1/(2q))/\varphi(q),\ell_q) ℓ q ∗ = max ( A q ( 1/ ( 2 q )) / φ ( q ) , ℓ q ) q ∈ E q\in\mathcal{E} q ∈ E ℓ q ∗ < c q \ell_q^*<c_q ℓ q ∗ < c q ∣ G ( θ ∗ ) ∣ = 2 n exp ( π ( N ) ( − log 2 + c q ) + o ( π ( N ) ) ) |G(\theta^*)|=2^n\exp(\pi(N)(-\log2+c_q)+o(\pi(N))) ∣ G ( θ ∗ ) ∣ = 2 n exp ( π ( N ) ( − log 2 + c q ) + o ( π ( N ))) q ∈ E q\in\mathcal{E} q ∈ E ≤ e − γ q π ( N ) ∣ G ( θ ∗ ) ∣ \le e^{-\gamma_q\pi(N)}|G(\theta^*)| ≤ e − γ q π ( N ) ∣ G ( θ ∗ ) ∣ γ q = ( c q − ℓ q ∗ ) / 4 > 0 \gamma_q=(c_q-\ell_q^*)/4>0 γ q = ( c q − ℓ q ∗ ) /4 > 0 φ ( z ) / s ∗ \varphi(z)/s^* φ ( z ) / s ∗ ∣ z ∣ ≤ log N |z|\le\sqrt{\log N} ∣ z ∣ ≤ log N V ( a / q ) = ∅ \mathcal{V}(a/q)=\varnothing V ( a / q ) = ∅ c q − ℓ q ∗ c_q-\ell_q^* c q − ℓ q ∗ q ≤ P q\le\mathcal{P} q ≤ P q = 6 q=6 q = 6 q ∉ E q\notin\mathcal{E} q ∈ / E A q ( 1 / ( 2 q ) ) ≤ φ ( q ) c q − 0 ≤ ℓ q A_q(1/(2q))\le\varphi(q)c_q-0\le\ell_q A q ( 1/ ( 2 q )) ≤ φ ( q ) c q − 0 ≤ ℓ q A q ( X ) A_q(X) A q ( X ) ≤ κ / X \le\kappa/X ≤ κ / X A q ( X ) → 0 A_q(X)\to0 A q ( X ) → 0 X → ∞ X\to\infty X → ∞ ℓ q ∗ = ℓ q \ell_q^*=\ell_q ℓ q ∗ = ℓ q
Proof.
(1) By Lemma M, ∑ p log ∣ cos π p θ ∣ ≤ ∑ p K M ( p θ ) = π ( N ) K ^ M ( 0 ) + ∑ 1 ≤ ∣ m ∣ ≤ M K ^ M ( m ) ∑ p ≤ N e ( m p θ ) \sum_p \log|\cos\pi p\theta| \le\sum_p K_M(p\theta)=\pi(N)\widehat{K}_M(0)+\sum_{1\le|m|\le M}\widehat{K}_M(m)\sum_{p\le N}e(mp\theta) ∑ p log ∣ cos π pθ ∣ ≤ ∑ p K M ( pθ ) = π ( N ) K M ( 0 ) + ∑ 1 ≤ ∣ m ∣ ≤ M K M ( m ) ∑ p ≤ N e ( m pθ ) K ^ M ( 0 ) ≤ − log 2 + C M − 1 / 2 \widehat{K}_M(0)\le-\log2+CM^{-1/2} K M ( 0 ) ≤ − log 2 + C M − 1/2 ∣ K ^ M ( m ) ∣ ≤ C |\widehat{K}_M(m)|\le C ∣ K M ( m ) ∣ ≤ C
If k ≤ P / M k\le\mathfrak{P}/M k ≤ P / M ∣ θ − b / ( m k ) ∣ ≤ P / ( m k N M ) ≤ P / ( m k N ) |\theta-b/(mk)|\le\mathfrak{P}/(mkNM)\le\mathfrak{P}/(mkN) ∣ θ − b / ( mk ) ∣ ≤ P / ( mk N M ) ≤ P / ( mk N ) m k ≤ P mk\le\mathfrak{P} mk ≤ P b / ( m k ) b/(mk) b / ( mk ) a ′ / q ′ a'/q' a ′ / q ′ q ′ ≤ m k q'\le mk q ′ ≤ mk θ \theta θ M ( a ′ / q ′ ) \mathfrak{M}(a'/q') M ( a ′ / q ′ ) P / M < k ≤ N M / P \mathfrak{P}/M<k\le NM/\mathfrak{P} P / M < k ≤ N M / P
Vinogradov–Vaughan [IK, Thm 13.6] then gives ∑ p ≤ N e ( m p θ ) ≪ N ( M / P ) 1 / 2 log 4 N \sum_{p\le N}e(mp\theta)\ll N(M/\mathfrak{P})^{1/2}\log^4N ∑ p ≤ N e ( m pθ ) ≪ N ( M / P ) 1/2 log 4 N N 4 / 5 log 4 N N^{4/5}\log^4N N 4/5 log 4 N O ( N log N ) O(\sqrt{N}\log N) O ( N log N ) log p \log p log p
(3) As in (2), split at ∣ X ∣ = N − 1 / 2 |X|=N^{-1/2} ∣ X ∣ = N − 1/2 0 _0 0 ∣ X ∣ ≤ N − 1 / 2 |X|\le N^{-1/2} ∣ X ∣ ≤ N − 1/2 q ∉ C q\notin\mathcal{C} q ∈ / C c q = 0 c_q=0 c q = 0 log ∣ F N ∗ ( a / q ) ∣ = n log 2 − π ( N ) log 2 + o ( π ( N ) ) \log|F_N^*(a/q)|=n\log2-\pi(N)\log2+o(\pi(N)) log ∣ F N ∗ ( a / q ) ∣ = n log 2 − π ( N ) log 2 + o ( π ( N )) ∣ X ∣ ≥ N − 1 / 2 |X|\ge N^{-1/2} ∣ X ∣ ≥ N − 1/2
Lemma L applies with A q ( ∣ X ∣ ) / φ ( q ) ≤ ℓ q ∗ A_q(|X|)/\varphi(q) \le\ell_q^* A q ( ∣ X ∣ ) / φ ( q ) ≤ ℓ q ∗ ∣ X ∣ ≥ 1 / ( 2 q ) |X| \ge1/(2q) ∣ X ∣ ≥ 1/ ( 2 q ) A q ( ∣ X ∣ ) < A q ( 0 ) = 0 ≤ φ ( q ) ℓ q ∗ A_q(|X|) < A_q(0) = 0 \le\varphi(q)\ell_q^* A q ( ∣ X ∣ ) < A q ( 0 ) = 0 ≤ φ ( q ) ℓ q ∗ N − 1 / 2 ≤ ∣ X ∣ < 1 / ( 2 q ) N^{-1/2} \le|X| < 1/(2q) N − 1/2 ≤ ∣ X ∣ < 1/ ( 2 q ) ℓ q ∗ ≥ 0 \ell_q^* \ge0 ℓ q ∗ ≥ 0 A q ( X ) → 0 A_q(X) \to0 A q ( X ) → 0 X → ∞ X \to\infty X → ∞
Remark (the formula of Proposition C). If ν a / q ≪ N 3 / 2 + ε / 2 \nu_{a/q} \ll N^{3/2+\varepsilon/2} ν a / q ≪ N 3/2 + ε /2 q q q q q q a / q a/q a / q e − ν σ e^{-\nu\sigma} e − ν σ a / q a/q a / q θ ∗ \theta^* θ ∗ A a / q \mathcal{A}_{a/q} A a / q V ( a / q ) = ∅ \mathcal{V}(a/q)=\varnothing V ( a / q ) = ∅ O ( N − 1 / 2 + 3 ε ) O(N^{-1/2+3\varepsilon}) O ( N − 1/2 + 3 ε ) ∣ Re z ∣ ≤ log N |\operatorname{Re}z| \le\sqrt{\log N} ∣ Re z ∣ ≤ log N z z z ∣ Im z ∣ ≍ ∣ ν ∣ / s ∗ |\operatorname{Im}z| \asymp|\nu|/s^* ∣ Im z ∣ ≍ ∣ ν ∣/ s ∗ V ( a / q ) = { w } \mathcal{V}(a/q)=\{w\} V ( a / q ) = { w } a / q a/q a / q s ~ 2 \tilde{s}^2 s ~ 2 G ~ \widetilde{G} G a / q a/q a / q T T T δ > 0 \delta>0 δ > 0
∣ R a / q ( T ) − A a / q ( T ) ∣ ≤ ∣ G ( a / q ) ∣ ( w e ν 2 / ( 2 s ~ 2 ) N − 3 / 2 + 7 ε s ~ 2 + w e − N 2 ε / ( 16384 log N ) + 4 e 4 ν 2 / S 2 e − N 2 ε / ( 2048 log N ) ) + 2 ⋅ 2 ( − log 2 + ℓ q ∗ + δ ) π ( N |R_{a/q}(T)-\mathcal{A}_{a/q}(T)| \le|G(a/q)|\left(\frac{we^{\nu^2/(2\tilde{s}^2)}N^{-3/2+7\varepsilon}}{\tilde{s}^2}+we^{-N^{2\varepsilon}/(16384\log N)}+4e^{4\nu^2/S_2}e^{-N^{2\varepsilon}/(2048\log N)}\right)+2\cdot2^{(-\log2+\ell_q^*+\delta)\pi(N} ∣ R a / q ( T ) − A a / q ( T ) ∣ ≤ ∣ G ( a / q ) ∣ ( s ~ 2 w e ν 2 / ( 2 s ~ 2 ) N − 3/2 + 7 ε + w e − N 2 ε / ( 16384 l o g N ) + 4 e 4 ν 2 / S 2 e − N 2 ε / ( 2048 l o g N ) ) + 2 ⋅ 2 ( − l o g 2 + ℓ q ∗ + δ ) π ( N for N ≥ N 0 ( δ ) N \ge N_0(\delta) N ≥ N 0 ( δ ) w ⋅ H ^ ′ ≈ − w ⋅ z φ ( z ) / s 2 w\cdot\widehat{H}'\approx-w\cdot z\varphi(z)/s^2 w ⋅ H ′ ≈ − w ⋅ z φ ( z ) / s 2 z = 0 z=0 z = 0 θ ∗ \theta^* θ ∗ ν a / q \nu_{a/q} ν a / q ≪ N 2 e − c log N \ll N^{2}e^{-c\sqrt{\log N}} ≪ N 2 e − c l o g N
The argument uses no hypothesis on the zeros. It is stated arc by arc. A single expansion “all arcs plus an exponentially small error” is not available pointwise in T T T Table 3 ); they are leading and exact in the residue-class counts.
Table 3. Relative error of the estimates of r N ( T ) r_N(T) r N ( T ) T = S N / 2 + k σ T=S_N/2+k\sigma T = S N /2 + k σ
The count itself: what the Gaussian already gives
Figure 3. Relative error of the estimates of r N ( T ) r_N(T) r N ( T ) W N = { 1 } ∪ { p ≤ 2000 } W_N=\{1\}\cup\{p\le2000\} W N = { 1 } ∪ { p ≤ 2000 } ∣ T − S / 2 ∣ ≤ 4.2 σ |T-S/2|\le4.2\sigma ∣ T − S /2∣ ≤ 4.2 σ 2 ⋅ 10 − 3 2\cdot10^{-3} 2 ⋅ 1 0 − 3 10 − 1 10^{-1} 1 0 − 1 4 σ 4\sigma 4 σ 1 / n 1/n 1/ n Table 3 ; the method of BENCH5, problem P4), about 10 − 10 10^{-10} 1 0 − 10 10 − 6 10^{-6} 1 0 − 6 4 σ 4\sigma 4 σ
The positivity of the count does not constrain the races. The arcs enter r N ( T ) r_N(T) r N ( T ) q = 6 q=6 q = 6 e − 0.144 π ( N ) e^{-0.144\pi(N)} e − 0.144 π ( N ) O ( N β / log N ) O(N^\beta/\log N) O ( N β / log N ) π ( N ) \pi(N) π ( N ) β = 1 \beta=1 β = 1
Prime races on the arcs
The characters seen by an arc
Expanding in Dirichlet characters, H a / q ( N ) = ∑ χ ≠ χ 0 A χ π ( N , χ ) H_{a/q}(N)=\sum_{\chi\ne\chi_0}A_\chi\pi(N,\chi) H a / q ( N ) = ∑ χ = χ 0 A χ π ( N , χ ) A χ = φ ( q ) − 1 ∑ r c r χ ‾ ( r ) A_\chi=\varphi(q)^{-1}\sum_r c_r\overline{\chi}(r) A χ = φ ( q ) − 1 ∑ r c r χ ( r ) ∑ r c r = 0 \sum_r c_r=0 ∑ r c r = 0 c − r = c r c_{-r}=c_r c − r = c r X ( a / q ) = { χ ≠ χ 0 : A χ ≠ 0 } X(a/q)=\{\chi\ne\chi_0:A_\chi\ne0\} X ( a / q ) = { χ = χ 0 : A χ = 0 }
A one-sided criterion
Proposition 3 (Landau’s theorem on an arc). Fix q q q a a a ( a , q ) = 1 (a,q)=1 ( a , q ) = 1
( R ) (R) ( R ) L ( σ , χ ) ≠ 0 L(\sigma,\chi)\ne0 L ( σ , χ ) = 0 σ ∈ ( 1 2 , 1 ] \sigma\in\left(\frac{1}{2},1\right] σ ∈ ( 2 1 , 1 ] χ ∈ X ( a / q ) \chi\in X(a/q) χ ∈ X ( a / q )
( N ) (N) ( N ) ρ \rho ρ Re ρ > 1 2 \operatorname{Re}\rho>\frac{1}{2} Re ρ > 2 1 ∑ χ ∈ X A χ m χ ( ρ ) ≠ 0 \sum_{\chi\in X}A_{\chi}m_{\chi}(\rho)\ne0 ∑ χ ∈ X A χ m χ ( ρ ) = 0 m χ ( ρ ) ≠ 0 m_{\chi}(\rho)\ne0 m χ ( ρ ) = 0 m χ ( ρ ) m_{\chi}(\rho) m χ ( ρ ) ρ \rho ρ L ( s , χ ) L(s,\chi) L ( s , χ )
Then the following are equivalent:
every L ( s , χ ) L(s,\chi) L ( s , χ ) χ ∈ X ( a / q ) \chi\in X(a/q) χ ∈ X ( a / q ) 0 < Re s < 1 0<\operatorname{Re}s<1 0 < Re s < 1 Re s = 1 2 \operatorname{Re}s=\frac{1}{2} Re s = 2 1
for every ε > 0 \varepsilon>0 ε > 0 C ε C_{\varepsilon} C ε H a / q ( N ) ≤ C ε N 1 2 + ε H_{a/q}(N)\le C_{\varepsilon}N^{\frac{1}{2}+\varepsilon} H a / q ( N ) ≤ C ε N 2 1 + ε N ≥ 2 N\ge2 N ≥ 2
for every ε > 0 \varepsilon>0 ε > 0 C ε C_{\varepsilon} C ε H a / q ( N ) ≥ − C ε N 1 2 + ε H_{a/q}(N)\ge-C_{\varepsilon}N^{\frac{1}{2}+\varepsilon} H a / q ( N ) ≥ − C ε N 2 1 + ε N ≥ 2 N\ge2 N ≥ 2
If (2) holds for one ε \varepsilon ε L ( s , χ ) L(s,\chi) L ( s , χ ) χ ∈ X \chi\in X χ ∈ X β ≤ 1 2 + ε \beta\le\frac{1}{2}+\varepsilon β ≤ 2 1 + ε
Proof. (1) ⇒ \Rightarrow ⇒ π ( N , χ ) = O ( N log N ) \pi(N,\chi)=O(\sqrt{N}\log N) π ( N , χ ) = O ( N log N ) H = O ( N log N ) H=O(\sqrt{N}\log N) H = O ( N log N )
(2) ⇒ \Rightarrow ⇒ θ = 1 2 + ε \theta=\frac{1}{2}+\varepsilon θ = 2 1 + ε D ( s ) = ∑ p ∤ q c p m o d q p − s D(s)=\sum_{p\nmid q}c_p\mathop{\rm mod}q p^{-s} D ( s ) = ∑ p ∤ q c p mod q p − s D ( s ) = ∑ χ ∈ X A χ ∑ k ≥ 1 μ ( k ) k − 1 log L ( k s , χ k ) = ∑ χ ∈ X A χ log L ( s , χ ) + h ( s ) D(s)=\sum_{\chi\in X}A_{\chi}\sum_{k\ge1}\mu(k)k^{-1}\log L(ks,\chi^{k})=\sum_{\chi\in X}A_{\chi}\log L(s,\chi)+h(s) D ( s ) = ∑ χ ∈ X A χ ∑ k ≥ 1 μ ( k ) k − 1 log L ( k s , χ k ) = ∑ χ ∈ X A χ log L ( s , χ ) + h ( s ) h h h Re s > 1 2 \operatorname{Re}s>\frac{1}{2} Re s > 2 1 L ( 1 , χ ) ≠ 0 L(1,\chi)\ne0 L ( 1 , χ ) = 0 D D D s = 1 s=1 s = 1 ( θ , 1 ] (\theta,1] ( θ , 1 ] G ( x ) = C ε x θ − H ( x ) ≥ 0 G(x)=C_{\varepsilon}x^{\theta}-H(x)\ge0 G ( x ) = C ε x θ − H ( x ) ≥ 0 ∫ 1 ∞ G ( x ) x − s − 1 d x = C ε / ( s − θ ) − D ( s ) / s \int_{1}^{\infty}G(x)x^{-s-1}\,dx=C_{\varepsilon}/(s-\theta)-D(s)/s ∫ 1 ∞ G ( x ) x − s − 1 d x = C ε / ( s − θ ) − D ( s ) / s Re s > 1 \operatorname{Re}s>1 Re s > 1 ≤ θ \le\theta ≤ θ D D D Re s > θ \operatorname{Re}s>\theta Re s > θ ρ \rho ρ L ( s , χ ) L(s,\chi) L ( s , χ ) Re ρ > θ \operatorname{Re}\rho>\theta Re ρ > θ D D D ∑ χ A χ m χ ( ρ ) \sum_{\chi}A_{\chi}m_{\chi}(\rho) ∑ χ A χ m χ ( ρ ) β ≤ 1 2 + ε \beta\le\frac{1}{2}+\varepsilon β ≤ 2 1 + ε ε \varepsilon ε χ ∗ \chi^{*} χ ∗ χ \chi χ β ≥ 1 2 \beta\ge\frac{1}{2} β ≥ 2 1 L ( s , χ ) = L ( s , χ ∗ ) ∏ p ∣ q ( 1 − χ ∗ ( p ) p − s ) L(s,\chi)=L(s,\chi^{*})\prod_{p\mid q}(1-\chi^{*}(p)p^{-s}) L ( s , χ ) = L ( s , χ ∗ ) ∏ p ∣ q ( 1 − χ ∗ ( p ) p − s ) Re s > 0 \operatorname{Re}s>0 Re s > 0 ⇒ \Rightarrow ⇒ G = C ε x θ + H G=C_{\varepsilon}x^{\theta}+H G = C ε x θ + H
The hypotheses are verified for q = 8 , 10 , 14 q=8,10,14 q = 8 , 10 , 14
( N ) (N) ( N ) X X X q = 8 , 10 q=8,10 q = 8 , 10 q = 14 q=14 q = 14 X = { χ , χ ˉ } X=\{\chi,\bar{\chi}\} X = { χ , χ ˉ } A χ ˉ = A ˉ χ A_{\bar{\chi}}=\bar{A}_{\chi} A χ ˉ = A ˉ χ A χ ( a ) = χ ( a ) A χ ( 1 ) A_{\chi}(a)=\chi(a)A_{\chi}(1) A χ ( a ) = χ ( a ) A χ ( 1 ) χ ( a ) \chi(a) χ ( a ) a = 1 a=1 a = 1 A χ / A ˉ χ = 0.0101 − 0.99995 i A_{\chi}/\bar{A}_{\chi}=0.0101-0.99995i A χ / A ˉ χ = 0.0101 − 0.99995 i Re A χ ( a ) ∈ { 0.1717 , 0.0614 , − 0.2331 } \operatorname{Re}A_{\chi}(a)\in\{0.1717,0.0614,-0.2331\} Re A χ ( a ) ∈ { 0.1717 , 0.0614 , − 0.2331 } A χ ( a ) / A ˉ χ ( a ) A_{\chi}(a)/\bar{A}_{\chi}(a) A χ ( a ) / A ˉ χ ( a ) L ( s , χ ˉ ) L(s,\bar{\chi}) L ( s , χ ˉ ) L ( s , χ ) L(s,\chi) L ( s , χ )
( R ) (R) ( R ) L ( s , χ ) = q − s ∑ a χ ( a ) Z ( s , a / q ) L(s,\chi)=q^{-s}\sum_a\chi(a)Z(s,a/q) L ( s , χ ) = q − s ∑ a χ ( a ) Z ( s , a / q ) Z Z Z [ 1 2 , 1 − 2 − 8 ] \left[\frac{1}{2},1-2^{-8}\right] [ 2 1 , 1 − 2 − 8 ] [ 1 − 2 − 8 , 1 ] [1-2^{-8},1] [ 1 − 2 − 8 , 1 ] ∣ s − 1 ∣ = 1 4 |s-1|=\frac{1}{4} ∣ s − 1∣ = 4 1 Table 4 lists the primitive characters; (R) for the characters mod 10 and 14 follows because 1 − χ ∗ ( 2 ) 2 − σ ≠ 0 1-\chi^{*}(2)2^{-\sigma}\ne0 1 − χ ∗ ( 2 ) 2 − σ = 0 σ > 0 \sigma>0 σ > 0 χ ˉ \bar{\chi} χ ˉ L ( σ , χ ˉ ) L(\sigma,\bar{\chi}) L ( σ , χ ˉ ) L ( σ , χ ) L(\sigma,\chi) L ( σ , χ )
Table 4.
Table 4 (PDF p. 18).
Table 4 (PDF p. 19). Certification of ( R ) \mathit{(R)} ( R ) < 227 < 227 < 227 10 4 10^{4} 1 0 4 ≤ 13 \le13 ≤ 13
For q = 8 q = 8 q = 8 10 10 10 14 14 14 ⇔ \Leftrightarrow ⇔ ⇔ \Leftrightarrow ⇔ a / q a/q a / q 2 π ( N ) e π ( N ) w ‾ ( q ) ⋅ O q ( 1 ) 2^{\pi(N)}e^{\pi(N)\overline{w}(q)}\cdot O_q(1) 2 π ( N ) e π ( N ) w ( q ) ⋅ O q ( 1 ) π ( N ) \pi(N) π ( N ) e C ε N 1 2 + ε e^{C_{\varepsilon}N^{\frac{1}{2}+\varepsilon}} e C ε N 2 1 + ε π ( N ) \pi(N) π ( N ) li N \operatorname{li}N li N π ( N ) − li N \pi(N)-\operatorname{li}N π ( N ) − li N ζ \zeta ζ
The case of weights ± 1 \pm1 ± 1 log ∣ cos ( π r a / q ) ∣ \log|\cos(\pi r a/q)| log ∣ cos ( π r a / q ) ∣
No last sign change
Theorem S. For every a a a 14 14 14 H a / 14 ( N ) = Ω ± ( N 1 2 − ε ) H_{a/14}(N) = \Omega_{\pm}(N^{\frac{1}{2}-\varepsilon}) H a /14 ( N ) = Ω ± ( N 2 1 − ε ) ε > 0 \varepsilon> 0 ε > 0
Proof. X ( a / 14 ) = { χ , χ ‾ } X(a/14)=\{\chi,\overline{\chi}\} X ( a /14 ) = { χ , χ } G ( x ) = C x θ − H a / 14 ( x ) ≥ 0 G(x)=Cx^{\theta}-H_{a/14}(x)\ge0 G ( x ) = C x θ − H a /14 ( x ) ≥ 0 x ≥ x 0 x\ge x_{0} x ≥ x 0 θ < 1 2 \theta<\frac{1}{2} θ < 2 1 G G G C / ( s − θ ) − D ( s ) / s + C/(s-\theta)-D(s)/s+ C / ( s − θ ) − D ( s ) / s + D ( s ) = ∑ χ ∈ X A χ ∑ k μ ( k ) k − 1 log L ( k s , χ k ) D(s)=\sum_{\chi\in X}A_{\chi}\sum_{k}\mu(k)k^{-1}\log L(ks,\chi^{k}) D ( s ) = ∑ χ ∈ X A χ ∑ k μ ( k ) k − 1 log L ( k s , χ k )
k = 1 k=1 k = 1 L ( s , χ ) L(s,\chi) L ( s , χ ) [ 1 2 , 1 ] [\frac{1}{2},1] [ 2 1 , 1 ] [ 1 3 , 1 2 ) [\frac{1}{3},\frac{1}{2}) [ 3 1 , 2 1 ) L ( s , χ ‾ ) L(s,\overline{\chi}) L ( s , χ ) ( 1 2 , 2 3 ] (\frac{1}{2},\frac{2}{3}] ( 2 1 , 3 2 ]
k = 2 k=2 k = 2 χ 2 = χ ‾ \chi^{2}=\overline{\chi} χ 2 = χ s > 1 3 s>\frac{1}{3} s > 3 1
k = 3 k=3 k = 3 χ 3 = χ 0 \chi^{3}=\chi_{0} χ 3 = χ 0 s = 1 3 s=\frac{1}{3} s = 3 1
k ≥ 4 k\ge4 k ≥ 4 Re s > 1 4 \operatorname{Re}s>\frac{1}{4} Re s > 4 1
So the first real singularity is at max ( θ , 1 3 ) < 1 2 \max(\theta,\frac{1}{3})<\frac{1}{2} max ( θ , 3 1 ) < 2 1 D D D Re s > max ( θ , 1 3 ) \operatorname{Re}s>\max(\theta,\frac{1}{3}) Re s > max ( θ , 3 1 ) L ( s , χ ) L(s,\chi) L ( s , χ ) ρ \rho ρ Re s = 1 2 \operatorname{Re}s=\frac{1}{2} Re s = 2 1 Im ρ ∈ [ 4.355 , 4.357 ] \operatorname{Im}\rho\in[4.355,4.357] Im ρ ∈ [ 4.355 , 4.357 ] D D D A χ m χ ( ρ ) + A χ ‾ m χ ‾ ( ρ ) A_{\chi}m_{\chi}(\rho)+A_{\overline{\chi}}m_{\overline{\chi}}(\rho) A χ m χ ( ρ ) + A χ m χ ( ρ ) k = 2 k=2 k = 2 L ( 2 ρ , χ ‾ ) ≠ 0 L(2\rho,\overline{\chi})\ne0 L ( 2 ρ , χ ) = 0 Re s = 1 \operatorname{Re}s=1 Re s = 1 A χ / A χ ‾ A_{\chi}/A_{\overline{\chi}} A χ / A χ ρ \rho ρ a a a A χ ( a ) = χ ( a ) A χ ( 1 ) A_{\chi}(a)=\chi(a)A_{\chi}(1) A χ ( a ) = χ ( a ) A χ ( 1 ) A χ / A χ ‾ A_{\chi}/A_{\overline{\chi}} A χ / A χ χ ( a ) 2 \chi(a)^{2} χ ( a ) 2 Re A χ ( a ) \operatorname{Re}A_{\chi}(a) Re A χ ( a )
For q = 8 q=8 q = 8 10 10 10 χ 2 = χ 0 \chi^{2}=\chi_{0} χ 2 = χ 0 s = 1 2 s=\frac{1}{2} s = 2 1
The principal arc carries ζ \zeta ζ
The heights see only non-principal characters because the mean of the weights was subtracted. ζ \zeta ζ π ( N ) ( log 2 + w ˉ ( q ) ) \pi(N)(\log2+\bar{w}(q)) π ( N ) ( log 2 + w ˉ ( q )) π ( N ) − li N \pi(N)-\operatorname{li}N π ( N ) − li N q = 1 q=1 q = 1
Proposition 30 (centre and width of the prime subset sums). For a uniformly random subset of the primes ≤ N \le N ≤ N T = ∑ x p p T=\sum x_p p T = ∑ x p p 1 2 M 1 ( N ) \frac{1}{2}M_1(N) 2 1 M 1 ( N ) 1 4 M 2 ( N ) \frac{1}{4}M_2(N) 4 1 M 2 ( N ) M k ( N ) = ∑ p ≤ N p k M_k(N)=\sum_{p\le N}p^k M k ( N ) = ∑ p ≤ N p k E k ( N ) = M k ( N ) − li ( N k + 1 ) E_k(N)=M_k(N)-\operatorname{li}(N^{k+1}) E k ( N ) = M k ( N ) − li ( N k + 1 ) li ( N k + 1 ) = ∫ N t k d t / log t \operatorname{li}(N^{k+1})=\int^N t^k\,dt/\log t li ( N k + 1 ) = ∫ N t k d t / log t k ≥ 0 k\ge0 k ≥ 0
the Riemann hypothesis;
for every ε > 0 \varepsilon>0 ε > 0 C ε C_\varepsilon C ε E k ( N ) ≤ C ε N k + 1 2 + ε E_k(N)\le C_\varepsilon N^{k+\frac{1}{2}+\varepsilon} E k ( N ) ≤ C ε N k + 2 1 + ε N ≥ 2 N\ge2 N ≥ 2
for every ε > 0 \varepsilon>0 ε > 0 C ε C_\varepsilon C ε E k ( N ) ≥ − C ε N k + 1 2 + ε E_k(N)\ge-C_\varepsilon N^{k+\frac{1}{2}+\varepsilon} E k ( N ) ≥ − C ε N k + 2 1 + ε
Proof. (1) ⇒ \Rightarrow ⇒ E k ( N ) = ∫ N t k d ( π − li ) ( t ) = N k ( π − li ) ( N ) − k ∫ N t k − 1 ( π − li ) ( t ) d t = O ( N k + 1 2 log N ) E_k(N)=\int^N t^k\,d(\pi-\operatorname{li})(t)=N^k(\pi-\operatorname{li})(N)-k\int^N t^{k-1}(\pi-\operatorname{li})(t)\,dt=O(N^{k+\frac{1}{2}}\log N) E k ( N ) = ∫ N t k d ( π − li ) ( t ) = N k ( π − li ) ( N ) − k ∫ N t k − 1 ( π − li ) ( t ) d t = O ( N k + 2 1 log N ) π ( N ) − li N = O ( N log N ) \pi(N)-\operatorname{li}N=O(\sqrt{N}\log N) π ( N ) − li N = O ( N log N ) ⇒ \Rightarrow ⇒ E k E_k E k
1 s [ log ( ( s − k − 1 ) ζ ( s − k ) ) + ∑ j ≥ 2 μ ( j ) j log ζ ( j ( s − k ) ) ] + ( entire ) , \frac{1}{s}\left[\log\left((s-k-1)\zeta(s-k)\right)+\sum_{j\ge2}\frac{\mu(j)}{j}\log\zeta\left(j(s-k)\right)\right]+(\text{entire}), s 1 [ log ( ( s − k − 1 ) ζ ( s − k ) ) + j ≥ 2 ∑ j μ ( j ) log ζ ( j ( s − k ) ) ] + ( entire ) , using P ( w ) = ∑ p p − w = ∑ j μ ( j ) j − 1 log ζ ( j w ) P(w)=\sum_p p^{-w}=\sum_j\mu(j)j^{-1}\log\zeta(jw) P ( w ) = ∑ p p − w = ∑ j μ ( j ) j − 1 log ζ ( j w ) ∫ ∞ t k − s d t / log t = − log ( s − k − 1 ) + ( entire ) \int^\infty t^{k-s}\,dt/\log t=-\log(s-k-1)+(\text{entire}) ∫ ∞ t k − s d t / log t = − log ( s − k − 1 ) + ( entire ) j ≥ 2 j\ge2 j ≥ 2 Re s > k + 1 2 \operatorname{Re}s>k+\frac{1}{2} Re s > k + 2 1 ζ ( σ ) < 0 \zeta(\sigma)<0 ζ ( σ ) < 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) ( k + 1 2 + ε , ∞ ) (k+\frac{1}{2}+\varepsilon,\infty) ( k + 2 1 + ε , ∞ ) C ε x k + 1 2 + ε − E k ( x ) ≥ 0 C_\varepsilon x^{k+\frac{1}{2}+\varepsilon}-E_k(x)\ge0 C ε x k + 2 1 + ε − E k ( x ) ≥ 0 log ζ ( s − k ) \log\zeta(s-k) log ζ ( s − k ) Re s > k + 1 2 + ε \operatorname{Re}s>k+\frac{1}{2}+\varepsilon Re s > k + 2 1 + ε Re ρ > 1 2 + ε \operatorname{Re}\rho>\frac{1}{2}+\varepsilon Re ρ > 2 1 + ε m ( ρ ) ≠ 0 m(\rho)\ne0 m ( ρ ) = 0 ζ \zeta ζ Re s > 1 2 \operatorname{Re}s>\frac{1}{2} Re s > 2 1 ⇒ \Rightarrow ⇒ C ε x k + 1 2 + ε + E k ( x ) ≥ 0 C_\varepsilon x^{k+\frac{1}{2}+\varepsilon}+E_k(x)\ge0 C ε x k + 2 1 + ε + E k ( x ) ≥ 0 C ε C_\varepsilon C ε 1 / log 2 1/\log2 1/ log 2 li ( x k + 1 ) \operatorname{li}(x^{k+1}) li ( x k + 1 )
The case k = 0 k=0 k = 0 π ( N ) − li N \pi(N)-\operatorname{li}N π ( N ) − li N k = 1 k=1 k = 1 k = 2 k=2 k = 2 li ( N 3 ) / 2 \sqrt{\operatorname{li}(N^3)}/2 li ( N 3 ) /2 1 + O ( N − 1 2 + ε ) 1+O(N^{-\frac{1}{2}+\varepsilon}) 1 + O ( N − 2 1 + ε ) 10 8 10^8 1 0 8 E 2 ( N ) / N 5 / 2 E_2(N)/N^{5/2} E 2 ( N ) / N 5/2 − 0.128 -0.128 − 0.128 − 0.0017 -0.0017 − 0.0017 N N N 10 2 10^2 1 0 2 10 8 10^8 1 0 8 π ( N ) < li N \pi(N)<\operatorname{li}N π ( N ) < li N
Figure 4 shows the three cases. It is a computation in a finite range and proves nothing about RH.
Figure 4. E k ( N ) / N k + 1 / 2 E_k(N)/N^{k+1/2} E k ( N ) / N k + 1/2 k = 0 k=0 k = 0 π ( N ) − li N \pi(N)-\operatorname{li}N π ( N ) − li N k = 1 k=1 k = 1 k = 2 k=2 k = 2 N N N 10 2 10^2 1 0 2 10 8 10^8 1 0 8 Proposition 30 , RH is equivalent, for each k k k E k ( N ) ≤ C ε N k + 1 / 2 + ε E_k(N) \le C_\varepsilon N^{k+1/2+\varepsilon} E k ( N ) ≤ C ε N k + 1/2 + ε k = 2 k=2 k = 2 − 0.128 -0.128 − 0.128 − 0.0017 -0.0017 − 0.0017
Proposition 30 is a reformulation, not a route to a proof: controlling these main terms at square-root scale is the error term of the prime number theorem.
The races measured
We computed H a / q ( N ) ⋅ log N / N H_{a/q}(N)\cdot\log N/\sqrt{N} H a / q ( N ) ⋅ log N / N N ∈ [ 10 4 , 10 8 ] N \in[10^4,10^8] N ∈ [ 1 0 4 , 1 0 8 ] μ = − φ ( q ) − 1 ∑ r c r # { b : b 2 ≡ r ( m o d q ) } \mu=-\varphi(q)^{-1}\sum_r c_r\#\{b:b^2\equiv r\pmod q\} μ = − φ ( q ) − 1 ∑ r c r # { b : b 2 ≡ r ( mod q )} ∑ χ ∣ A χ ∣ 2 ∑ γ ( 1 4 + γ 2 ) − 1 \sum_\chi|A_\chi|^2\sum_\gamma(\frac{1}{4}+\gamma^2)^{-1} ∑ χ ∣ A χ ∣ 2 ∑ γ ( 4 1 + γ 2 ) − 1 L ( 1 , χ ) L(1,\chi) L ( 1 , χ ) L ′ ( 1 , χ ) L'(1,\chi) L ′ ( 1 , χ )
The arcs as an algorithm
What changes for arbitrary weights
For an arbitrary set W W W o ( W ) > 1 o(W)>1 o ( W ) > 1
In the dense regime (2 n ≫ S 2^n \gg S 2 n ≫ S
The arcs see congruences: by Remark 3.1 the class counts N c ( q ) = # { S ⊆ W : ∑ S ≡ c ( m o d q ) } N_c(q)=\#\{S\subseteq W:\sum S\equiv c\pmod q\} N c ( q ) = # { S ⊆ W : ∑ S ≡ c ( mod q )} F F F q q q O ( n q ) O(nq) O ( n q )
A decider and a constructor
Decider. It has two parts: residue obstructions from the exact N c ( q ) N_c(q) N c ( q ) q ≤ 40 q \le40 q ≤ 40 q = 1 q = 1 q = 1
Table 6. The arc decider. The obstructions are found automatically by the arcs; all errors are in the tails, where the Gaussian threshold is coarse.
Constructor by self-reduction. Weights are sorted decreasingly. At step k k k w k w_k w k T − w k T-w_k T − w k w k w_k w k − ∞ -\infty − ∞
With arcs only, down to the last weight, the success rate falls from 88% to 8% as n n n a / ( S k + 1 ) a/(S_k+1) a / ( S k + 1 )
Why sixteen weights: an empirical Poisson rule
As a working model, the number of subsets of the remaining block hitting T T T λ ( T ) = 2 m G ( T ; S m / 2 , σ m 2 ) ⋅ ( class factor ) \lambda(T) = 2^mG(T;S_m/2,\sigma_m^2)\cdot(\text{class factor}) λ ( T ) = 2 m G ( T ; S m /2 , σ m 2 ) ⋅ ( class factor ) λ \lambda λ ≈ e − λ \approx e^{-\lambda} ≈ e − λ Pr ( success ) ≈ exp ( − ∑ k e − λ k ) \Pr(\text{success}) \approx\exp(-\sum_k e^{-\lambda_k}) Pr ( success ) ≈ exp ( − ∑ k e − λ k )
Table 7. Empty chosen branch, about 1.3 ⋅ 10 6 1.3\cdot10^6 1.3 ⋅ 1 0 6 The model is accurate near λ = 4 \lambda= 4 λ = 4 λ = 7.6 \lambda= 7.6 λ = 7.6 § 8.6 shows that the true tail of the endgame count decays like a power of λ \lambda λ δ \delta δ n n n 2 m / 2 π σ m 2^m/\sqrt{2\pi\sigma_m} 2 m / 2 π σ m λ ∗ = ln ( n / δ ) \lambda^* = \ln(n/\delta) λ ∗ = ln ( n / δ ) 2 m ∗ = λ ∗ 2 π σ m ∗ 2^{m^*} = \lambda^*\sqrt{2\pi\sigma_{m^*}} 2 m ∗ = λ ∗ 2 π σ m ∗ m ∗ ≈ 15 m^* \approx15 m ∗ ≈ 15
With the filtered endgame of §5.6 a larger block costs little more, since the cost of a round grows only like ( λ ∗ ) 1 / 4 (\lambda^*)^{1/4} ( λ ∗ ) 1/4 ( λ ∗ = 5 ⋅ 10 4 (\lambda^* = 5 \cdot10^4 ( λ ∗ = 5 ⋅ 1 0 4
Dummies, blur and de-blurring
The solver reads the arcs directly on the original weights. This subsection explains why that reading is exact, through an augmentation that the solver itself never performs.
A simple augmentation restores the floor: insert dummy weights d = 1 + S d = 1+S d = 1 + S S S S o ≤ 1 o \le1 o ≤ 1 K = 1 K = 1 K = 1 1 , 2 , … , 2 k − 1 1, 2, \ldots, 2^{k-1} 1 , 2 , … , 2 k − 1 min W \min W min W
Blur identity. In that case
F W ∪ D ( z ) = F W ( z ) ∏ d ∈ D ( 1 + z d ) = F W ( z ) 1 − z M 1 − z , M = 2 k F_{W\cup D}(z) = F_W(z) \prod_{d\in D}(1+z^d) = F_W(z)\frac{1-z^M}{1-z}, \qquad M=2^k F W ∪ D ( z ) = F W ( z ) d ∈ D ∏ ( 1 + z d ) = F W ( z ) 1 − z 1 − z M , M = 2 k so the dummies blur the counts with a flat window of width M M M o o o
r W ( T ) − r W ( T − M ) = r W ∪ D ( T ) − r W ∪ D ( T − 1 ) . r_W(T)-r_W(T-M)=r_{W\cup D}(T)-r_{W\cup D}(T-1). r W ( T ) − r W ( T − M ) = r W ∪ D ( T ) − r W ∪ D ( T − 1 ) . Lemma 5.1 (gap lemma). Every interval of length Σ D + 1 \Sigma_D+1 Σ D + 1 [ 0 , S ] [0,S] [ 0 , S ] W W W
Proof. Every point T T T s W + s D s_W+s_D s W + s D s D ≤ Σ D s_D \le\Sigma_D s D ≤ Σ D W ∪ D W\cup D W ∪ D
De-blurring the arcs. The arcs are traced on W ∪ D W\cup D W ∪ D W W W W ∪ D W\cup D W ∪ D N c W ( q ) N_c^W(q) N c W ( q ) N c W ∪ D ( q ) N_c^{W\cup D}(q) N c W ∪ D ( q ) ( 1 + x d m o d q ) (1+x^{d\bmod q}) ( 1 + x d mod q ) Z [ x ] / ( x q − 1 ) \mathbb{Z}[x]/(x^q-1) Z [ x ] / ( x q − 1 ) q / gcd ( q , d ) q/\gcd(q,d) q / g cd( q , d ) 1 + z 1+z 1 + z z = − 1 z=-1 z = − 1 1 2 \frac{1}{2} 2 1 1 + x 1+x 1 + x Z [ x ] / ( x q − 1 ) \mathbb{Z}[x]/(x^q-1) Z [ x ] / ( x q − 1 ) q q q
Three findings fix the design.
The de-blurred scores coincide exactly with the scores of the arcs on the original weights: HURS with dummies plus de-blurred arcs, and arcs on the originals, are one algorithm seen from two sides.
Without de-blurring (arcs on W ∪ D W\cup D W ∪ D M M M
Marking the dummies (weights M w M w M w M d + 1 M d + 1 M d + 1 M T M T M T q = M q=M q = M W ∪ D W\cup D W ∪ D
Class counts in floating point. Since the de-blurred scores coincide with the scores on the original weights, the implementation computes the suffix class counts directly on W W W W ∪ D W\cup D W ∪ D T > ∑ W + ∑ D T>\sum W+\sum D T > ∑ W + ∑ D ( q , c ) (q,c) ( q , c )
The recurrence N c ← N c + N c − w N_c \leftarrow N_c+N_{c-w} N c ← N c + N c − w n ⋅ ε n\cdot\varepsilon n ⋅ ε
Each row is rescaled by a power of two. A positive count lies in [ 1 , 2 n ] [1,2^n] [ 1 , 2 n ] n ≤ 1020 n\leq1020 n ≤ 1020 2 − 1020 2^{-1020} 2 − 1020
The Fourier side would be diagonal but unstable, because the inverse transform cancels down to the size of the minor arcs. The class side is stable.
Against the exact integer counts there are 0 discrepancies in the zero pattern and ∣ Δ s c o r e ∣ ≤ 3 ⋅ 10 − 14 |\Delta\mathrm{score}|\leq3\cdot10^{-14} ∣Δ score ∣ ≤ 3 ⋅ 1 0 − 14 O ( n Q 2 ) O(nQ^2) O ( n Q 2 ) n = 200 n=200 n = 200
When o ≤ 1 o\leq1 o ≤ 1 O ( n ) O(n) O ( n ) { 1 } ∪ { p < 3000 } \{1\}\cup\{p<3000\} { 1 } ∪ { p < 3000 }
Cost law and benchmarks
Cost law, first version. On generic sets σ m ∗ \sigma_{m}^{*} σ m ∗ [ 0 , min W ) [0,\min W) [ 0 , min W ) o ≈ min W o \approx\min W o ≈ min W m ∗ ≈ log 2 o + log 2 λ ∗ + c m^{*} \approx\log_{2} o+\log_{2}\lambda^{*}+c m ∗ ≈ log 2 o + log 2 λ ∗ + c
T ( n , o ) = O ( n Q ) + O ( ρ ∗ o poly ( m ∗ ) ) . T(n,o)=O(nQ)+O\left(\sqrt{\rho^{*}o}\,\operatorname{poly}(m^{*})\right). T ( n , o ) = O ( n Q ) + O ( ρ ∗ o poly ( m ∗ ) ) . The time ratio between o = 1.4 ⋅ 10 10 o=1.4\cdot10^{10} o = 1.4 ⋅ 1 0 10 o = 2 ⋅ 10 7 o=2\cdot10^{7} o = 2 ⋅ 1 0 7 700 = 26 \sqrt{700}=26 700 = 26
Cost law, filtered endgame. With P ≈ λ / 2 ≈ λ ∗ / 2 P\approx\lambda/2\approx\lambda^{*}/2 P ≈ λ /2 ≈ λ ∗ /2 2 m ≈ λ ∗ 2 π σ m 2^{m}\approx\lambda^{*}\sqrt{2\pi}\sigma_{m} 2 m ≈ λ ∗ 2 π σ m O ( m ( λ ∗ σ m ) 1 / 4 + m ( σ m / λ ∗ ) 1 / 2 ) O\left(m(\lambda^{*}\sigma_{m})^{1/4}+m(\sigma_{m}/\lambda^{*})^{1/2}\right) O ( m ( λ ∗ σ m ) 1/4 + m ( σ m / λ ∗ ) 1/2 ) λ ∗ ≍ σ m 1 / 3 \lambda^{*}\asymp\sigma_{m}^{1/3} λ ∗ ≍ σ m 1/3 σ m 1 / 3 \sigma_{m}^{1/3} σ m 1/3
T ( n , o ) = O ( n Q ) + O ( o 1 / 3 poly ( m ) ) . T(n,o)=O(nQ)+O\left(o^{1/3}\operatorname{poly}(m)\right). T ( n , o ) = O ( n Q ) + O ( o 1/3 poly ( m ) ) . On the instances of Table 9 ′ the measured times (15–30 ms at large λ ∗ \lambda^{*} λ ∗ m = 40 m=40 m = 40 Table 9 ″).
Table 9. Benchmarks, first version. The first four rows use exact ground truth over 200 reachable targets. No wrong output occurred in more than 2400 runs. At 10 10 10^{10} 1 0 10 10 12 10^{12} 1 0 12 2 ⋅ 10 14 2\cdot10^{14} 2 ⋅ 1 0 14 2 ⋅ 10 16 2\cdot10^{16} 2 ⋅ 1 0 16 2 100 2^{100} 2 100 λ ∗ \lambda^{*} λ ∗
The endgame in the language of arcs. On a block B B B m m m m ⋅ R m\cdot R m ⋅ R R = ∑ B R=\sum_{B} R = ∑ B F B = F B 1 F B 2 F_{B}=F_{B_{1}}F_{B_{2}} F B = F B 1 F B 2 ∣ B 1 ∣ = ∣ B 2 ∣ = m / 2 \lvert B_{1}\rvert=\lvert B_{2}\rvert=m/2 ∣ B 1 ∣ = ∣ B 2 ∣ = m /2 m ⋅ 2 m / 2 m\cdot2^{m/2} m ⋅ 2 m /2 § 7.6 restricts each half-spectrum to one residue class mod P P P L < R L<R L < R o \sqrt{o} o
What the arcs contribute: an ablation.
Where the algorithm and the races meet. On primes restricted to a progression (q = 3 q = 3 q = 3 p ≡ 1 ( m o d 4 ) p \equiv1 \pmod{4} p ≡ 1 ( mod 4 ) 1 / 8 1/8 1/8 3 / 8 3/8 3/8 log ∣ F ∣ \log|F| log ∣ F ∣ π ( 4000 ; 8 , 5 ) − π ( 4000 ; 8 , 1 ) \pi(4000;8,5)-\pi(4000;8,1) π ( 4000 ; 8 , 5 ) − π ( 4000 ; 8 , 1 )
Density one. We write ρ = n / log 2 D \rho= n/\log_2 D ρ = n / log 2 D D = max W − min W + 1 D = \max W-\min W+1 D = max W − min W + 1 S ≈ n ⋅ max W / 2 S \approx n\cdot\max W/2 S ≈ n ⋅ max W /2 2 n / S ≥ 1 2^n/S \ge1 2 n / S ≥ 1 ρ ≳ 1 + log 2 ( n / 2 ) / log 2 max W \rho\gtrsim1+\log_2(n/2)/\log_2\max W ρ ≳ 1 + log 2 ( n /2 ) / log 2 max W 2 n / S ≈ 1 2^n/S \approx1 2 n / S ≈ 1 ≈ 2 n \approx2^n ≈ 2 n 2 n / 2 2^{n/2} 2 n /2 λ ≈ 1 \lambda\approx1 λ ≈ 1 § 7.6 has nothing to remove. There the arcs carry no usable information. The largest arc over q ≤ 24 q \le24 q ≤ 24 − 0.30 -0.30 − 0.30 − 0.43 -0.43 − 0.43 e − 0.3 n e^{-0.3n} e − 0.3 n e − 0.43 n e^{-0.43n} e − 0.43 n ≈ 2 0.39 n \approx2^{0.39n} ≈ 2 0.39 n 2 0.337 n 2^{0.337n} 2 0.337 n 2 0.291 n 2^{0.291n} 2 0.291 n u = W − 1 ( m o d N ) u=W^{-1}\pmod N u = W − 1 ( mod N ) u / N u/N u / N q ∼ N q\sim N q ∼ N
The filtered endgame
The endgame block E E E m m m λ \lambda λ § 7.3 ), and the plain meet-in-the-middle enumerates all 2 m / 2 2^{m/2} 2 m /2
The filter. Split E E E P P P r r r ≡ r ( m o d P ) \equiv r \pmod P ≡ r ( mod P ) ≡ t − r \equiv t-r ≡ t − r 1 / P 1/P 1/ P 1 / P 2 1/P^2 1/ P 2
Parameters. The descent hands over when the peak density falls below λ ∗ = 5 ⋅ 10 4 \lambda^* = 5\cdot10^4 λ ∗ = 5 ⋅ 1 0 4 λ ≥ 8 \lambda\ge8 λ ≥ 8 m ≥ 16 m \ge16 m ≥ 16 § 7.7 , Parameters derived from the instance ).
The prime. P P P [ P 0 , 2 P 0 ) [P_0,2P_0) [ P 0 , 2 P 0 ) P 0 = max ( 3 , ⌈ λ / 2 ⌉ ) P_0=\max(3,\lceil\lambda/2\rceil) P 0 = max ( 3 , ⌈ λ /2 ⌉) [ P 0 , 2 P 0 ) [P_0,2P_0) [ P 0 , 2 P 0 ) P 0 ≥ 100 P_0\ge100 P 0 ≥ 100 ( 1 − o ( 1 ) ) / ( 6 ln m M ) (1-o(1))/(6\ln mM) ( 1 − o ( 1 )) / ( 6 ln m M ) m m m P 0 P_0 P 0 λ ∗ \lambda^* λ ∗ § 8.7 ). The uniform draw is the one the theorem assumes. The runs of Table 9 ′–9″ predate it and used sympy.randprime, which returns the next prime after a uniform integer and so favours the primes that follow long gaps. Rerun with the uniform draw, Table 10 ′ gives identical outcomes, and the reference run BENCH5 v2 of § 7.7 uses it.
Table 10. On random weights the arcs are not needed: steering to the centre plus the exact endgame achieves the cost law. On sets with an almost divisor the arcs are essential: the class counts see the obstruction mod d d d
Rounds and fallback. The implementation draws a fresh prime and residue for up to six rounds, then falls back to the full meet-in-the-middle if m ≤ 44 m\le44 m ≤ 44 m > 44 m>44 m > 44 2 22 2^{22} 2 22 λ < 8 \lambda<8 λ < 8
correlated through its weights; Theorem R″ uses disjoint blocks, so its independence argument does not cover them.
Theorem mode. A second mode runs the algorithm that Theorem R″ analyses: R R R m m m b = 30 b = 30 b = 30 34 34 34 m = 42 m = 42 m = 42 47 47 47 P 0 = 436 P_0 = 436 P 0 = 436 825 825 825 R = 6 R = 6 R = 6 b = 30 b = 30 b = 30 p ≈ 1 / 164 p \approx1/164 p ≈ 1/164 g = 3 g = 3 g = 3 R = 6 R = 6 R = 6 R R R 300 ⋅ ln ( 1 / δ ) 300 \cdot\ln(1/\delta) 300 ⋅ ln ( 1/ δ ) n ≳ 1.2 ⋅ 10 4 ⋅ ln ( 1 / δ ) n \gtrsim1.2 \cdot10^4 \cdot\ln(1/\delta) n ≳ 1.2 ⋅ 1 0 4 ⋅ ln ( 1/ δ )
In the language of arcs. The list { y ⊆ A : S y ≡ r mod P } \{y \subseteq A : S_y \equiv r \mathbin{\operatorname{mod}} P\} { y ⊆ A : S y ≡ r mod P } r r r F A F_A F A P P P [ F A ] ≡ r ⋅ [ F B ] ≡ t − r [F_A]_{\equiv r} \cdot[F_B]_{\equiv t-r} [ F A ] ≡ r ⋅ [ F B ] ≡ t − r F B 1 F B 2 F_{B_1}F_{B_2} F B 1 F B 2 § 7.5 . The probability that a round succeeds is the fraction of live classes at the modulus P P P § 8.7 ), the same quantity that corrects the decider of § 7.2 , here at a modulus of the order of λ \lambda λ P P P
Merging. Sorted half-lists are built by merging L L L L + w L + w L + w O ( 2 h ) O(2^h) O ( 2 h ) O ( h ⋅ 2 h ) O(h \cdot2^h) O ( h ⋅ 2 h )
Table 9′. Time per target on the instances and targets of Table 9 (rows 5–7, same seed), first version against the filtered endgame, on one machine and one core. Every cell found 20/20 targets with 0 wrong outputs and 0 fallbacks. The first version needs 4.4 s at 10 12 10^{12} 1 0 12
Table 9″. The endgame alone, 20 blocks per row, targets within M / 2 M/2 M /2 λ \lambda λ P 0 = λ / 2 < 100 P_0 = \lambda/2 < 100 P 0 = λ /2 < 100
Dynamic moduli
The fixed moduli q = 2 , … , 24 q = 2, \ldots, 24 q = 2 , … , 24
The rule. Let W W W N c ( k ) ( q ) N_c^{(k)}(q) N c ( k ) ( q ) W k = ( w k , … , w n ) W_k = (w_k,\ldots,w_n) W k = ( w k , … , w n ) m = n − k + 1 m = n-k+1 m = n − k + 1 ≡ c ( m o d q ) \equiv c \pmod q ≡ c ( mod q )
δ q ( k ) = max c ∣ q N c ( k ) ( q ) 2 m − 1 ∣ , ε k = min ( 1 , 1 / m ) . \delta_q(k) = \max_c \left|\frac{qN_c^{(k)}(q)}{2^m}-1\right|,\qquad\varepsilon_k = \min(1,1/m). δ q ( k ) = c max 2 m q N c ( k ) ( q ) − 1 , ε k = min ( 1 , 1/ m ) . Candidates. C = { 2 , … , 24 } ∪ { d ≤ 2000 : d ∣ gcd ( a , b ) for one of 64 pairs ( a , b ) of weights drawn at random } \mathcal{C} = \{2,\ldots,24\} \cup\{d \le2000 : d \mid\gcd(a,b)\text{ for one of 64 pairs }(a,b)\text{ of weights drawn at random}\} C = { 2 , … , 24 } ∪ { d ≤ 2000 : d ∣ g cd( a , b ) for one of 64 pairs ( a , b ) of weights drawn at random } d d d f f f 1 − ( 1 − f ) 2 1-(1-f)^2 1 − ( 1 − f ) 2 d d d ( 1 − ( 1 − f ) 2 ) 64 (1-(1-f)^2)^{64} ( 1 − ( 1 − f ) 2 ) 64 10 − 80 10^{-80} 1 0 − 80 f = 5 / 200 f = 5/200 f = 5/200
Selection. At suffix k k k Q ( k ) = { q ∈ C : δ q ( k ) ≥ ε k } Q(k) = \{q \in\mathcal{C} : \delta_q(k) \ge\varepsilon_k\} Q ( k ) = { q ∈ C : δ q ( k ) ≥ ε k } max ( 0 , max q ∈ Q ( k ) ln ( q N c ( k ) ( q ) / 2 m ) ) \max(0,\max_{q\in Q(k)}\ln(qN_c^{(k)}(q)/2^m)) max ( 0 , max q ∈ Q ( k ) ln ( q N c ( k ) ( q ) / 2 m )) c = t m o d q c = t \bmod q c = t mod q − ∞ -\infty − ∞ N T m o d q ( 1 ) ( q ) = 0 N_{T\bmod q}^{(1)}(q) = 0 N T mod q ( 1 ) ( q ) = 0 q ∈ Q ( 1 ) q \in Q(1) q ∈ Q ( 1 ) ( q , T m o d q ) (q,T\bmod q) ( q , T mod q ) ( 0 , 0 ) (0,0) ( 0 , 0 ) T < 0 T < 0 T < 0 T > ∑ W T > \sum W T > ∑ W ( q , T m o d q ) (q,T\bmod q) ( q , T mod q )
when that class is empty mod q q q ( − 3 , − 3 ) (-3,-3) ( − 3 , − 3 ) T T T
Cost. The divisors of the 64 gcds cost O ( w max ) O(\sqrt{w_{\max}}) O ( w m a x ) O ( n ⋅ ∑ q ∈ C q ) O(n \cdot\sum_{q\in\mathcal{C}}q) O ( n ⋅ ∑ q ∈ C q ) § 7.4 , and the descent costs O ( n + ∑ k ∣ Q ( k ) ∣ ) O(n+\sum_k |Q(k)|) O ( n + ∑ k ∣ Q ( k ) ∣ )
Lemma 5.3 (pruning). Fix k k k t t t s C s_{\mathcal{C}} s C s Q s_Q s Q § 7.2 with the moduli C \mathcal{C} C Q ( k ) Q(k) Q ( k )
If N t ( k ) m o d q = 0 N_t^{(k)} \bmod q=0 N t ( k ) mod q = 0 q ∈ C q\in\mathcal{C} q ∈ C q ∈ Q ( k ) q\in Q(k) q ∈ Q ( k ) s Q = s C = − ∞ s_Q=s_{\mathcal{C}}=-\infty s Q = s C = − ∞ Q ( 1 ) Q(1) Q ( 1 ) C \mathcal{C} C
Otherwise, if 0 ≤ t ≤ S k 0\le t\le S_k 0 ≤ t ≤ S k − ∞ -\infty − ∞ 0 ≤ s C − s Q ≤ ln ( 1 + ε k ) ≤ ε k 0\le s_{\mathcal{C}}-s_Q\le\ln(1+\varepsilon_k)\le\varepsilon_k 0 ≤ s C − s Q ≤ ln ( 1 + ε k ) ≤ ε k
By Lemma C(1), applied to the suffix, δ q ( k ) ≤ ∑ d ∣ q , d ′ > 1 φ ( d ′ ) e − π 2 κ k ( d ′ ) / ( 2 d ′ 2 ) \delta_q(k)\le\sum_{d\mid q,d'>1}\varphi(d')e^{-\pi^2\kappa_k(d')/(2d'^2)} δ q ( k ) ≤ ∑ d ∣ q , d ′ > 1 φ ( d ′ ) e − π 2 κ k ( d ′ ) / ( 2 d ′2 ) κ k ( d ′ ) \kappa_k(d') κ k ( d ′ ) W k W_k W k d ′ d' d ′ q ∉ Q ( k ) q\notin Q(k) q ∈ / Q ( k ) κ k ( d ′ ) > ( 2 d ′ 2 / π 2 ) ln ( ( q − 1 ) / ε k ) \kappa_k(d')>(2d'^2/\pi^2)\ln((q-1)/\varepsilon_k) κ k ( d ′ ) > ( 2 d ′2 / π 2 ) ln (( q − 1 ) / ε k ) d ′ > 1 d'>1 d ′ > 1 q q q
Proof.
An empty class gives ∣ q ⋅ 0 / 2 m − 1 ∣ = 1 ≥ ε k |q\cdot0/2^m-1|=1\ge\varepsilon_k ∣ q ⋅ 0/ 2 m − 1∣ = 1 ≥ ε k δ q ( k ) ≥ ε k \delta_q(k)\ge\varepsilon_k δ q ( k ) ≥ ε k q ∈ Q ( k ) q\in Q(k) q ∈ Q ( k ) − ∞ -\infty − ∞ q q q k = 1 k=1 k = 1
The class t m o d q t\bmod q t mod q q ∈ C q\in\mathcal{C} q ∈ C q ∉ Q ( k ) q\notin Q(k) q ∈ / Q ( k ) δ q ( k ) < ε k \delta_q(k)<\varepsilon_k δ q ( k ) < ε k q N C / 2 m < 1 + ε k qN_{\mathcal{C}}/2^m<1+\varepsilon_k q N C / 2 m < 1 + ε k ln ( 1 + ε k ) \ln(1+\varepsilon_k) ln ( 1 + ε k ) b C = max ( b Q , max q ∉ Q ( k ) ln ( q N C / 2 m ) ) b_{\mathcal{C}}=\max(b_Q,\max_{q\notin Q(k)}\ln(qN_{\mathcal{C}}/2^m)) b C = max ( b Q , max q ∈ / Q ( k ) ln ( q N C / 2 m )) b Q b_Q b Q b Q ≥ 0 b_Q\ge0 b Q ≥ 0 b Q ≤ b C ≤ max ( b Q , ln ( 1 + ε k ) ) b_Q\le b_{\mathcal{C}}\le\max(b_Q,\ln(1+\varepsilon_k)) b Q ≤ b C ≤ max ( b Q , ln ( 1 + ε k ))
Lemma C(1) bounds δ q ( k ) \delta_q(k) δ q ( k ) ∑ d ∣ q , d ′ > 1 φ ( d ′ ) = q − 1 \sum_{d\mid q,d'>1}\varphi(d')=q-1 ∑ d ∣ q , d ′ > 1 φ ( d ′ ) = q − 1 e − π 2 κ k ( d ′ ) / ( 2 d ′ 2 ) < ε k / ( q − 1 ) e^{-\pi^2\kappa_k(d')/(2d'^2)}<\varepsilon_k/(q-1) e − π 2 κ k ( d ′ ) / ( 2 d ′2 ) < ε k / ( q − 1 ) ε k \varepsilon_k ε k
So the dynamic rule changes no NO answer and moves a score by at most ln ( 1 + ε k ) \ln(1+\varepsilon_k) ln ( 1 + ε k ) ln ( 1 + ε k + 1 ) \ln(1+\varepsilon_{k+1}) ln ( 1 + ε k + 1 ) λ = e s \lambda=e^s λ = e s P 0 P_0 P 0 1 + ε k 1+\varepsilon_k 1 + ε k δ q ( k ) \delta_q(k) δ q ( k ) Figure 7 shows both cases.
Figure 7. The selection rule of Lemma 5.3 along the suffixes (suffix size m m m δ 1009 \delta_{1009} δ 1009 q = 1009 q = 1009 q = 1009 q ≤ 24 q \le24 q ≤ 24 ε = min ( 1 , 1 / m ) \varepsilon= \min(1,1/m) ε = min ( 1 , 1/ m ) [ 1 , 10 10 ] [1,10^{10}] [ 1 , 1 0 10 ] 10 − 16 10^{-16} 1 0 − 16
Table 10′ (PDF p. 31). Fixed against dynamic moduli , 20 planted targets per row, one machine and one core; time per target where all 20 were found. The last column lists the smallest moduli read at the suffixes of more than 40 weights, and their mean number per suffix. There were 0 wrong outputs in every cell. On random weights the dynamic rule reads no modulus and costs what the centre rule costs. The moduli 101 and 1009 are larger than 24, and only the gcd candidates find them. With five non-divisible weights at most 2 5 = 32 2^5 = 32 2 5 = 32 { 1 } ∪ { p < 3000 } \{1\} \cup\{p < 3000\} { 1 } ∪ { p < 3000 }
Figure 8 follows one descent on an instance of this kind. Most decisions at the large weights are nearly indifferent; five are forced by the arc mod 1009.
Figure 8. The descent of ARCS on one planted target of an instance with 195 multiples of 1009 and 5 other weights (n = 200 n = 200 n = 200 10 − 5 10^{-5} 1 0 − 5
A preregistered check. The same comparison is part of a benchmark of five problems whose criteria were fixed, and hashed, before any instance was generated. Instance i i i P P P s + 1000 P + i s + 1000P + i s + 1000 P + i s s s q q q § 7.6 . Problem P5 has 200 weights, 195 multiples of 1009 and 5 others, with 6 planted and 6 impossible targets. The dynamic solver found 6/6 planted targets, in a median time of 0.061 s, and certified all 6 impossible ones with the certificate q = 1009 q = 1009 q = 1009 Figure 9 gives the other problems, including the one that ARCS loses.
Figure 9. BENCH5 v2, the reference run (criterion sha256 f81f1037. . . ; one run; every answer checked by the independent checker). Bars: correct answers per method; labels: correct/total and the median time of the correct answers. P1: reconciliation of 800 amounts. P2: a change-free payment from 1000 amounts. P3: a perfect partition of 200 values. P5: the instances above. ARCS (blue) solves 6/6 in every panel and certifies all 6 impossible targets of P5 with q = 1009 q = 1009 q = 1009 { 1 } ∪ { p ≤ N } \{1\} \cup\{p \le N\} { 1 } ∪ { p ≤ N } ≤ 4.3 ⋅ 10 − 8 \le4.3 \cdot10^{-8} ≤ 4.3 ⋅ 1 0 − 8 2.7 ⋅ 10 4 2.7 \cdot10^{4} 2.7 ⋅ 1 0 4 1.2 ⋅ 10 5 1.2 \cdot10^{5} 1.2 ⋅ 1 0 5 N = 2000 N = 2000 N = 2000 N = 2000 N = 2000 N = 2000
Parameters derived from the instance. The default form of ARCS runs the steps of this section with every constant derived before the descent from the instance and from two user inputs: the accepted failure probability δ \delta δ m a x o p s \mathrm{max}_{\mathrm{ops}} max ops ρ = n / log 2 D \rho= n/\log_{2} D ρ = n / log 2 D V k V_k V k
λ ∗ = max ( 400 , ( 2.3 n / δ ) 1 / 3 ) \lambda^{*} = \max(400,(2.3n/\delta)^{1/3}) λ ∗ = max ( 400 , ( 2.3 n / δ ) 1/3 ) § 8.6 with a union bound over the n n n P 0 ≥ 100 P_0 \ge100 P 0 ≥ 100
the suffix k ∗ k^{*} k ∗ 2 m / ( 2 π σ m ) 2^{m}/(\sqrt{2\pi}\sigma_m) 2 m / ( 2 π σ m ) λ ∗ \lambda^{*} λ ∗ m ∗ = n − k ∗ m^{*} = n-k^{*} m ∗ = n − k ∗
R = ⌈ ln ( 1 / δ ) / 2 ⌉ R = \left\lceil\ln(1/\delta)/2 \right\rceil R = ⌈ ln ( 1/ δ ) /2 ⌉ λ \lambda λ 1 / P 1/P 1/ P
the moduli: q q q ∑ q ≤ n \sum q \le n ∑ q ≤ n ⌈ ln δ / ln ( 3 / 4 ) ⌉ \left\lceil\ln\delta/ \ln(3/4) \right\rceil ⌈ ln δ / ln ( 3/4 ) ⌉ ≤ δ \le\delta ≤ δ
The handover rule depends on V k V_k V k k ∗ k^{*} k ∗ Table 8 ). If the predicted cost or memory exceeds the limits, ARCS answers FAIL with the prediction, after its NO certificates. Randomness comes from the operating system, and every prime and residue drawn is part of the output, so a run is replayable and P P P
Table 8. Table 8. Cost of the solver; Q Q Q O ( n ⋅ ∑ q ∈ Q q ) O(n\cdot\sum_{q\in\mathcal{Q}}q) O ( n ⋅ ∑ q ∈ Q q ) O ( w max ) O(\sqrt{w_{\max}}) O ( w m a x ) O ( n + ∑ k ∣ Q ( k ) ∣ ) O(n+\sum_k|Q(k)|) O ( n + ∑ k ∣ Q ( k ) ∣ ) k ≤ ⌈ log 2 w max ⌉ k\leq\lceil\log_2w_{\max}\rceil k ≤ ⌈ log 2 w m a x ⌉ W ∪ D W\cup D W ∪ D O ( n 2 Q 2 / 64 ) O(n^2Q^2/64) O ( n 2 Q 2 /64 ) m ∗ m^* m ∗ O ( m ∗ ⋅ 2 m ∗ / 2 ) O(m^*\cdot2^{m^*/2}) O ( m ∗ ⋅ 2 m ∗ /2 )
A second preregistered benchmark, with the same rules (criterion hashed before the instances, one run, independent checker), ran the default solver alone, with δ = 10 − 3 \delta= 10^{-3} δ = 1 0 − 3 m a x o p s = 2 34 \mathrm{max}_{\mathrm{ops}} = 2^{34} max ops = 2 34 2 30 2^{30} 2 30
P1, P2, P3 and P5 of BENCH5, with new seeds: every instance answered, the impossible targets with the certificates q = 2 q = 2 q = 2 q = 1009 q = 1009 q = 1009
50 weights of 50 bits (ρ ≈ 1 \rho\approx1 ρ ≈ 1 § 7.6 refuse: 6/6 planted targets found and 6/6 random targets refused by an exhaustive endgame, verified by the checker’s own meet-in-the-middle, in 6–11 s;
40 weights of 70 bits, beyond 64-bit integers: 6/6 found;
40 weights of 60 bits (ρ ≈ 0.67 \rho\approx0.67 ρ ≈ 0.67
100 weights of 110 bits (ρ ≈ 0.91 \rho\approx0.91 ρ ≈ 0.91 2 54 2^{54} 2 54 2 55.6 2^{55.6} 2 55.6
The predicted block equalled the actual one in all 54 runs that reached an endgame.
Rigorous versions of the algorithm
The outputs are verified, so correctness is never in question. What needs proof is that a solution is found, with high probability and within the stated cost. We give this for random weights (Theorems R and R′) and for dense structured sets (Corollary C′).
Setting. W = ( w 1 , … , w n ) W = (w_1,\ldots,w_n) W = ( w 1 , … , w n ) S = ∑ w i S = \sum w_i S = ∑ w i T T T H H H E E E ∣ E ∣ = m |E| = m ∣ E ∣ = m x i x_i x i H H H ∑ i ∈ E x i w i = t \sum_{i\in E} x_iw_i = t ∑ i ∈ E x i w i = t
The descent never fails
Lemma A (centre rule). Process H H H k k k R k R_k R k t k t_k t k c k = ( R k + S E ) / 2 c_k=(R_k+S_E)/2 c k = ( R k + S E ) /2 x k x_k x k ∣ t k − c k ∣ |t_k-c_k| ∣ t k − c k ∣ e k = t k − c k e_k=t_k-c_k e k = t k − c k w ∗ = max H w i w^*=\max_{H} w_i w ∗ = max H w i
∣ e k ∣ ≤ max ( ∣ e 0 ∣ − 1 2 ∑ j ≤ k w j , 1 2 max j ≤ k w j ) . |e_k| \le\max\left(|e_0|-\frac{1}{2}\sum_{j\le k}w_j,\frac{1}{2}\max_{j\le k}w_j\right). ∣ e k ∣ ≤ max ∣ e 0 ∣ − 2 1 j ≤ k ∑ w j , 2 1 j ≤ k max w j . In particular, if ∣ T − S / 2 ∣ ≤ S H / 2 + w ∗ / 2 |T-S/2|\le S_H/2+w^*/2 ∣ T − S /2∣ ≤ S H /2 + w ∗ /2 ∣ t − S E / 2 ∣ ≤ w ∗ / 2 |t-S_E/2|\le w^*/2 ∣ t − S E /2∣ ≤ w ∗ /2
Proof. c k = c k − 1 − w k / 2 c_k=c_{k-1}-w_k/2 c k = c k − 1 − w k /2 ∣ e k ∣ = ∣ ∣ e k − 1 ∣ − w k / 2 ∣ |e_k|=\left||e_{k-1}|-w_k/2\right| ∣ e k ∣ = ∣ ∣ e k − 1 ∣ − w k /2 ∣
Writing x i = ( 1 + ε i ) / 2 x_i=(1+\varepsilon_i)/2 x i = ( 1 + ε i ) /2 ∑ E ε i w i = u \sum_E\varepsilon_iw_i=u ∑ E ε i w i = u u = 2 t − S E u=2t-S_E u = 2 t − S E ∣ u ∣ ≤ w ∗ |u|\le w^* ∣ u ∣ ≤ w ∗
A local limit theorem uniform in the size of the weights
Lemma B0′. Let w 1 , w 2 , … w_1,w_2,\ldots w 1 , w 2 , … { 1 , … , M } \{1,\ldots,M\} { 1 , … , M } M ≥ 2 M\ge2 M ≥ 2 μ = ( M + 1 ) / 2 \mu=(M+1)/2 μ = ( M + 1 ) /2 σ 2 = ( M 2 − 1 ) / 12 \sigma^2=(M^2-1)/12 σ 2 = ( M 2 − 1 ) /12 κ 4 = − ( M 4 − 1 ) / 120 \kappa_4=-(M^4-1)/120 κ 4 = − ( M 4 − 1 ) /120 S k = w 1 + ⋯ + w k S_k=w_1+\cdots+w_k S k = w 1 + ⋯ + w k z = ( v − k μ ) / ( σ k ) z=(v-k\mu)/(\sigma\sqrt{k}) z = ( v − k μ ) / ( σ k ) g k ( v ) = ( 2 π k σ 2 ) − 1 / 2 e − z 2 / 2 g_k(v)=(2\pi k\sigma^2)^{-1/2}e^{-z^2/2} g k ( v ) = ( 2 π k σ 2 ) − 1/2 e − z 2 /2 q M = 1 / ( M sin ( π / ( 2 M ) ) ) ≤ 1 / 2 q_M=1/(M\sin(\pi/(2M)))\le1/\sqrt{2} q M = 1/ ( M sin ( π / ( 2 M ))) ≤ 1/ 2 k ≥ 2 k\ge2 k ≥ 2 v v v
∣ Pr ( S k = v ) − g k ( v ) ( 1 + κ 4 H e 4 ( z ) 24 σ 4 k ) ∣ ≤ 0.34 σ k 5 / 2 + 1 σ ( q M k − 2 12 + ( 0.41 k + 0.11 ) e − 0.308 k ) . \left|\Pr(S_k=v)-g_k(v)\left(1+\frac{\kappa_4He_4(z)}{24\sigma^4k}\right)\right|\le\frac{0.34}{\sigma k^{5/2}}+\frac{1}{\sigma}\left(\frac{q_M^{k-2}}{\sqrt{12}}+\left(\frac{0.41}{k}+0.11\right)e^{-0.308k}\right). Pr ( S k = v ) − g k ( v ) ( 1 + 24 σ 4 k κ 4 H e 4 ( z ) ) ≤ σ k 5/2 0.34 + σ 1 ( 12 q M k − 2 + ( k 0.41 + 0.11 ) e − 0.308 k ) . Consequently ∣ Pr ( S k = v ) − g k ( v ) ∣ ≤ C ( k , M ) / ( σ k 3 / 2 ) |\Pr(S_k=v)-g_k(v)|\le C(k,M)/(\sigma k^{3/2}) ∣ Pr ( S k = v ) − g k ( v ) ∣ ≤ C ( k , M ) / ( σ k 3/2 )
C ( k , M ) = 0.0599 M 2 + 1 M 2 − 1 + 0.34 k + k 3 / 2 ( q M k − 2 12 + ( 0.41 k + 0.11 ) e − 0.308 k ) . C(k,M)=0.0599\frac{M^2+1}{M^2-1}+\frac{0.34}{k}+k^{3/2}\left(\frac{q_M^{k-2}}{\sqrt{12}}+\left(\frac{0.41}{k}+0.11\right)e^{-0.308k}\right). C ( k , M ) = 0.0599 M 2 − 1 M 2 + 1 + k 0.34 + k 3/2 ( 12 q M k − 2 + ( k 0.41 + 0.11 ) e − 0.308 k ) . The constant 0.0599 = ( 1.2 / 24 ) ⋅ 3 / 2 π 0.0599=(1.2/24)\cdot3/\sqrt{2\pi} 0.0599 = ( 1.2/24 ) ⋅ 3/ 2 π z = 0 z=0 z = 0 ∣ κ 4 ∣ / σ 4 → 6 / 5 |\kappa_4|/\sigma^4\to6/5 ∣ κ 4 ∣/ σ 4 → 6/5
Proof. The characteristic function of w − μ w-\mu w − μ f ( θ ) = sin ( M θ / 2 ) / ( M sin ( θ / 2 ) ) f(\theta)=\sin(M\theta/2)/(M\sin(\theta/2)) f ( θ ) = sin ( M θ /2 ) / ( M sin ( θ /2 )) x = θ / 2 x=\theta/2 x = θ /2 y = M θ / 2 y=M\theta/2 y = M θ /2 Λ ( t ) = log ( sin t / t ) + t 2 / 6 \Lambda(t)=\log(\sin t/t)+t^2/6 Λ ( t ) = log ( sin t / t ) + t 2 /6 log f = − σ 2 θ 2 / 2 + Λ ( y ) − Λ ( x ) \log f=-\sigma^2\theta^2/2+\Lambda(y)-\Lambda(x) log f = − σ 2 θ 2 /2 + Λ ( y ) − Λ ( x )
On ∣ θ ∣ ≤ π / M |\theta|\le\pi/M ∣ θ ∣ ≤ π / M Λ ( t ) = − t 4 / 180 + Λ 6 ( t ) \Lambda(t)=-t^4/180+\Lambda_6(t) Λ ( t ) = − t 4 /180 + Λ 6 ( t ) ∣ Λ 6 ( t ) ∣ ≤ 4.35 ⋅ 10 − 4 t 6 |\Lambda_6(t)|\le4.35\cdot10^{-4}t^6 ∣ Λ 6 ( t ) ∣ ≤ 4.35 ⋅ 1 0 − 4 t 6 ( 0 , π / 2 ] (0,\pi/2] ( 0 , π /2 ] ( y 4 − x 4 ) / 180 = − κ 4 θ 4 / 24 (y^4-x^4)/180=-\kappa_4\theta^4/24 ( y 4 − x 4 ) /180 = − κ 4 θ 4 /24 f k = e − k σ 2 θ 2 / 2 ( 1 + k κ 4 θ 4 / 24 + E ) f^k=e^{-k\sigma^2\theta^2/2}(1+k\kappa_4\theta^4/24+E) f k = e − k σ 2 θ 2 /2 ( 1 + k κ 4 θ 4 /24 + E ) ∣ E ∣ ≤ ( k κ 4 θ 4 / 24 ) 2 / 2 + k ∣ Λ 6 ∣ e k ∣ Λ 6 ∣ |E|\le(k\kappa_4\theta^4/24)^2/2+k|\Lambda_6|e^{k|\Lambda_6|} ∣ E ∣ ≤ ( k κ 4 θ 4 /24 ) 2 /2 + k ∣ Λ 6 ∣ e k ∣ Λ 6 ∣ ≤ 0.146 / ( σ k 5 / 2 ) \le0.146/(\sigma k^{5/2}) ≤ 0.146/ ( σ k 5/2 ) ≤ 0.19 / ( σ k 5 / 2 ) \le0.19/(\sigma k^{5/2}) ≤ 0.19/ ( σ k 5/2 ) κ 4 2 / σ 8 ≤ 4 \kappa_4^2/\sigma^8\le4 κ 4 2 / σ 8 ≤ 4 ( M / σ ) 6 ≤ 4096 (M/\sigma)^6\le4096 ( M / σ ) 6 ≤ 4096
On π / M ≤ ∣ θ ∣ ≤ π \pi/M\le|\theta|\le\pi π / M ≤ ∣ θ ∣ ≤ π ∣ f ∣ ≤ q M |f|\le q_M ∣ f ∣ ≤ q M ( 2 π ) − 1 ∫ ∣ f ∣ 2 = 1 / M ≤ 1 / ( 12 σ ) (2\pi)^{-1}\int|f|^2=1/M\le1/(\sqrt{12}\sigma) ( 2 π ) − 1 ∫ ∣ f ∣ 2 = 1/ M ≤ 1/ ( 12 σ )
The Gaussian and Edgeworth tails beyond π / M \pi/M π / M ≤ ( 0.41 / k + 0.11 ) e − 0.308 k / σ \le(0.41/k+0.11)e^{-0.308k}/\sigma ≤ ( 0.41/ k + 0.11 ) e − 0.308 k / σ k σ 2 π 2 / ( 2 M 2 ) ≥ 0.308 k k\sigma^2\pi^2/(2M^2)\ge0.308k k σ 2 π 2 / ( 2 M 2 ) ≥ 0.308 k
Against exact convolutions (M ∈ { 2 , 3 , 5 , 16 , 64 , 317 } M\in\{2,3,5,16,64,317\} M ∈ { 2 , 3 , 5 , 16 , 64 , 317 } 2 ≤ k ≤ 160 2\le k\le160 2 ≤ k ≤ 160 C ( k ) C(k) C ( k ) M M M M = 2 M = 2 M = 2
The constant comes from the fourth cumulant. An envelope ∣ f ( θ ) ∣ ≤ 2 / ( M ∣ θ ∣ ) |f(\theta)| \le2/(M|\theta|) ∣ f ( θ ) ∣ ≤ 2/ ( M ∣ θ ∣ ) [ π / M , π ] [\pi/M,\pi] [ π / M , π ] ∣ f ∣ ⋅ M ∣ θ ∣ / 2 |f| \cdot M|\theta|/2 ∣ f ∣ ⋅ M ∣ θ ∣/2 π / 2 \pi/2 π /2 π / ( M ∣ θ ∣ ) \pi/(M|\theta|) π / ( M ∣ θ ∣ ) k k k
The endgame is non-empty
Lemma B. Let the weights of E E E { 1 , … , M } \{1,\ldots,M\} { 1 , … , M } ∣ E ∣ = m |E| = m ∣ E ∣ = m Z ( u ) = # { ϵ ∈ { ± 1 } E : ∑ ϵ i w i = u } Z(u) = \#\{\epsilon\in\{\pm1\}^{E} : \sum\epsilon_iw_i = u\} Z ( u ) = # { ϵ ∈ { ± 1 } E : ∑ ϵ i w i = u }
λ m = 2 m + 1 M 2 π m c M , c M = 1 3 + 1 2 M + 1 6 M 2 . \lambda_m = \frac{2^{m+1}}{M\sqrt{2\pi m c_M}}, \qquad c_M = \frac{1}{3}+\frac{1}{2M}+\frac{1}{6M^2}. λ m = M 2 π m c M 2 m + 1 , c M = 3 1 + 2 M 1 + 6 M 2 1 . Fix g > 0 g > 0 g > 0 η = 1 / 10 \eta= 1/10 η = 1/10 M ≥ 2 M \ge2 M ≥ 2 s s s ∣ s − m μ ∣ ≤ g σ m |s-m\mu| \le g\sigma\sqrt{m} ∣ s − m μ ∣ ≤ g σ m u u u ∣ u ∣ ≤ M |u| \le M ∣ u ∣ ≤ M u ≡ s ( m o d 2 ) u \equiv s \pmod{2} u ≡ s ( mod 2 )
λ : = E [ Z ( u ) ∣ S E = s ] = λ m ( 1 + o ( 1 ) ) \lambda:= E[Z(u)\mid S_E=s] = \lambda_m(1+o(1)) λ := E [ Z ( u ) ∣ S E = s ] = λ m ( 1 + o ( 1 )) F F F
E [ Z ( u ) 2 ∣ S E = s ] ≤ λ + λ ⋅ 1 [ u = 0 ] + λ 2 ( 1 + O ( m − 1 / 5 ) ) E[Z(u)^2\mid S_E=s] \le\lambda+\lambda\cdot1[u=0]+\lambda^2(1+O(m^{-1/5})) E [ Z ( u ) 2 ∣ S E = s ] ≤ λ + λ ⋅ 1 [ u = 0 ] + λ 2 ( 1 + O ( m − 1/5 ))
Pr ( Z ( u ) = 0 ∣ S E = s ) ≤ 2 / λ + O ( m − 1 / 5 ) \Pr(Z(u)=0\mid S_E=s) \le2/\lambda+O(m^{-1/5}) Pr ( Z ( u ) = 0 ∣ S E = s ) ≤ 2/ λ + O ( m − 1/5 )
For u ∈ { 0 , 1 } u\in\{0,1\} u ∈ { 0 , 1 } S E S_E S E
Proof. With ϵ i = + 1 \epsilon_i=+1 ϵ i = + 1 x ⊆ E x\subseteq E x ⊆ E ∑ ϵ i w i = 2 S x − S E \sum\epsilon_iw_i=2S_x-S_E ∑ ϵ i w i = 2 S x − S E x x x S x = a : = ( s + u ) / 2 S_x=a:=(s+u)/2 S x = a := ( s + u ) /2
(1) λ = ∑ x Pr ( S x = a ) Pr ( S E ∖ x = s − a ) / Pr ( S E = s ) \lambda=\sum_x\Pr(S_x=a)\Pr(S_{E\setminus x}=s-a)/\Pr(S_E=s) λ = ∑ x Pr ( S x = a ) Pr ( S E ∖ x = s − a ) / Pr ( S E = s ) ∣ k − m / 2 ∣ ≤ m 3 / 5 |k-m/2|\le m^{3/5} ∣ k − m /2∣ ≤ m 3/5 k = ∣ x ∣ k=|x| k = ∣ x ∣ x x x ≤ 2 e − 2 m 1 / 5 \le2e^{-2m^{1/5}} ≤ 2 e − 2 m 1/5 O ( m / M ) O(\sqrt{m}/M) O ( m / M ) ≤ 1 / M \le1/M ≤ 1/ M λ m \lambda_m λ m x x x O ( m − 1 / 2 ) O(m^{-1/2}) O ( m − 1/2 ) a a a S k S_k S k S m = s S_m=s S m = s N ( k s / m , σ 2 k ( m − k ) / m ) N(ks/m,\sigma^2k(m-k)/m) N ( k s / m , σ 2 k ( m − k ) / m ) σ 2 m / 4 \sigma^2m/4 σ 2 m /4 k = m / 2 + j k=m/2+j k = m /2 + j λ m ( 1 + o ( 1 ) ) \lambda_m(1+o(1)) λ m ( 1 + o ( 1 )) M M M ∣ u ∣ ≤ M |u|\le M ∣ u ∣ ≤ M
(2) For a pair ( x , y ) (x,y) ( x , y ) C = x ∩ y C=x\cap y C = x ∩ y A = x ∖ y A=x\setminus y A = x ∖ y B = y ∖ x B=y\setminus x B = y ∖ x D = E ∖ ( x ∪ y ) D=E\setminus(x\cup y) D = E ∖ ( x ∪ y ) S A = S B = a − S C S_A=S_B=a-S_C S A = S B = a − S C S D = S C − u S_D=S_C-u S D = S C − u
The diagonal gives λ \lambda λ A , B A,B A , B C C C D D D y = E ∖ x y=E\setminus x y = E ∖ x u = 0 u=0 u = 0 λ ⋅ 1 [ u = 0 ] \lambda\cdot1[u=0] λ ⋅ 1 [ u = 0 ]
For every other pair, bounding two factors by their maxima and ∑ c Pr ( S C = c ) Pr ( S D = c − u ) = Pr ( S C − S D = u ) \sum_c\Pr(S_C=c)\Pr(S_D=c-u)=\Pr(S_C-S_D=u) ∑ c Pr ( S C = c ) Pr ( S D = c − u ) = Pr ( S C − S D = u ) C m 3 / 2 Cm^{3/2} C m 3/2 e − c m 2 η e^{-cm^{2\eta}} e − c m 2 η
For typical pairs the four class sums S A S_A S A S B S_B S B S C S_C S C S D S_D S D ∣ A ∣ |A| ∣ A ∣ ∣ B ∣ |B| ∣ B ∣ ∣ C ∣ |C| ∣ C ∣ ∣ D ∣ |D| ∣ D ∣ M M M S E S_E S E
normal with covariance σ 2 ( ∑ ε i ε i ′ − ( ∑ ε i ) ( ∑ ε i ′ ) / m ) \sigma^2(\sum\varepsilon_i\varepsilon_i' - (\sum\varepsilon_i)(\sum\varepsilon_i')/m) σ 2 ( ∑ ε i ε i ′ − ( ∑ ε i ) ( ∑ ε i ′ ) / m ) r = O ( m − 1 / 2 + η ) r = O(m^{-1/2+\eta}) r = O ( m − 1/2 + η ) 1 + O ( m − 1 / 5 ) 1+O(m^{-1/5}) 1 + O ( m − 1/5 ) ∣ x ∣ |x| ∣ x ∣ ∣ y ∣ |y| ∣ y ∣ ∣ x △ y ∣ |x\triangle y| ∣ x △ y ∣ m 3 / 5 m^{3/5} m 3/5 m / 2 m/2 m /2 m / 8 m/8 m /8 c c c λ ≥ λ m / 2 \lambda\ge\lambda_m/2 λ ≥ λ m /2
Why the signed form, and the centre at S E / 2 S_E/2 S E /2 For a target fixed independently of E E E 1 / 1 − 1 4 ≈ 1.15 1/\sqrt{1-\frac{1}{4}} \approx1.15 1/ 1 − 4 1 ≈ 1.15 S E / 2 S_E/2 S E /2
Lemma B‴ (exact reduction to first moments). Let λ J ( v , t ) = E [ { ε ∈ { ± 1 } J : ∑ J ε i w i = v } ∣ S J = t ] \lambda_J(v,t) = E[\{\varepsilon\in\{\pm1\}^J : \sum_J \varepsilon_i w_i = v\}\mid S_J=t] λ J ( v , t ) = E [{ ε ∈ { ± 1 } J : ∑ J ε i w i = v } ∣ S J = t ]
E [ Z ( u ) 2 ∣ S E = s ] = ∑ k = 0 m ( m k ) ∑ s 1 Pr ( S m − k = s 1 ) Pr ( S k = s − s 1 ) Pr ( S m = s ) λ m − k ( u , s 1 ) λ k ( 0 , s − s 1 ) . E[Z(u)^2\mid S_E=s] = \sum_{k=0}^{m} \binom{m}{k}\sum_{s_1} \frac{\Pr(S_{m-k}=s_1)\Pr(S_k=s-s_1)}{\Pr(S_m=s)}\lambda_{m-k}(u,s_1)\lambda_k(0,s-s_1). E [ Z ( u ) 2 ∣ S E = s ] = k = 0 ∑ m ( k m ) s 1 ∑ Pr ( S m = s ) Pr ( S m − k = s 1 ) Pr ( S k = s − s 1 ) λ m − k ( u , s 1 ) λ k ( 0 , s − s 1 ) . Proof. Split a pair ( ε , ε ′ ) (\varepsilon,\varepsilon') ( ε , ε ′ ) m − k m-k m − k k k k ∑ ε w = ∑ ε ′ w = u \sum\varepsilon w=\sum\varepsilon' w=u ∑ εw = ∑ ε ′ w = u u u u 0 0 0
Proposition B′ (the constant of the second moment). Fix M ≥ 2 M \ge2 M ≥ 2 g > 0 g > 0 g > 0 z = ( s − m μ ) / ( σ m ) z=(s-m\mu)/(\sigma\sqrt{m}) z = ( s − m μ ) / ( σ m ) ∣ z ∣ ≤ g |z| \le g ∣ z ∣ ≤ g ∣ u ∣ ≤ M |u| \le M ∣ u ∣ ≤ M u ≡ s ( m o d 2 ) u \equiv s\pmod{2} u ≡ s ( mod 2 )
E [ Z 2 ∣ S E = s ] λ 2 = 1 + 1 + 1 [ u = 0 ] λ + κ m + γ M z m 3 / 2 + O M , g ( m − 2 ) , \frac{E[Z^2\mid S_E=s]}{\lambda^2}
=1+\frac{1+\mathbf{1}[u=0]}{\lambda}+\frac{\kappa}{m}+\frac{\gamma_M z}{m^{3/2}}+O_{M,g}(m^{-2}), λ 2 E [ Z 2 ∣ S E = s ] = 1 + λ 1 + 1 [ u = 0 ] + m κ + m 3/2 γ M z + O M , g ( m − 2 ) , and, conditioning only on the event S E ≡ u ( m o d 2 ) S_E\equiv u\pmod{2} S E ≡ u ( mod 2 ) Z ‾ u = E [ Z ∣ S E ≡ u ( m o d 2 ) ] \overline{Z}_u=E[Z\mid S_E\equiv u\pmod{2}] Z u = E [ Z ∣ S E ≡ u ( mod 2 )]
E [ Z 2 ∣ S E ≡ u ( m o d 2 ) ] Z ‾ u 2 = 1 + 1 + 1 [ u = 0 ] Z ‾ u + κ m + c 2 ( M , u ) m 2 + O M ( m − 3 ) , c 2 ( M , u ) = c 2 ( M , 0 ) − 2 κ u 2 E w 2 , \frac{E[Z^2\mid S_E\equiv u\pmod{2}]}{\overline{Z}_u^2}
=1+\frac{1+\mathbf{1}[u=0]}{\overline{Z}_u}+\frac{\kappa}{m}+\frac{c_2(M,u)}{m^2}+O_M(m^{-3}),\qquad c_2(M,u)=c_2(M,0)-\frac{2\kappa u^2}{Ew^2}, Z u 2 E [ Z 2 ∣ S E ≡ u ( mod 2 )] = 1 + Z u 1 + 1 [ u = 0 ] + m κ + m 2 c 2 ( M , u ) + O M ( m − 3 ) , c 2 ( M , u ) = c 2 ( M , 0 ) − E w 2 2 κ u 2 , with
κ = Var ( w 2 ) 4 ( E w 2 ) 2 → 1 5 , κ c = Var ( w 2 ) − Cov ( w 2 , w ) 2 / Var ( w ) 4 ( E w 2 ) 2 → 1 80 , \kappa=\frac{\operatorname{Var}(w^2)}{4(Ew^2)^2}\to\frac{1}{5},\qquad
\kappa_c=\frac{\operatorname{Var}(w^2)-\operatorname{Cov}(w^2,w)^2/\operatorname{Var}(w)}{4(Ew^2)^2}\to\frac{1}{80}, κ = 4 ( E w 2 ) 2 Var ( w 2 ) → 5 1 , κ c = 4 ( E w 2 ) 2 Var ( w 2 ) − Cov ( w 2 , w ) 2 / Var ( w ) → 80 1 , γ M = σ ( B − 2 β A ) 4 ( E w 2 ) 2 → − 3 80 , A = Var ( w 2 ) − Cov ( w 2 , w ) 2 Var ( w ) , β = Cov ( w , w 2 ) Var ( w ) E w 2 , B = E [ R 2 ( w − μ ) ] Var ( w ) , \gamma_M=\frac{\sigma(B-2\beta A)}{4(Ew^2)^2}\to-\frac{\sqrt{3}}{80},\qquad
A=\operatorname{Var}(w^2)-\frac{\operatorname{Cov}(w^2,w)^2}{\operatorname{Var}(w)},\quad
\beta=\frac{\operatorname{Cov}(w,w^2)}{\operatorname{Var}(w)Ew^2},\quad
B=\frac{E[R^2(w-\mu)]}{\operatorname{Var}(w)}, γ M = 4 ( E w 2 ) 2 σ ( B − 2 β A ) → − 80 3 , A = Var ( w 2 ) − Var ( w ) Cov ( w 2 , w ) 2 , β = Var ( w ) E w 2 Cov ( w , w 2 ) , B = Var ( w ) E [ R 2 ( w − μ )] , where R = w 2 − E w 2 − ( Cov ( w 2 , w ) / Var ( w ) ) ( w − μ ) R=w^2-Ew^2-(\operatorname{Cov}(w^2,w)/\operatorname{Var}(w))(w-\mu) R = w 2 − E w 2 − ( Cov ( w 2 , w ) / Var ( w )) ( w − μ )
Proof. Draw signs ε , ε ′ ∈ { ± 1 } E \varepsilon,\varepsilon'\in\{\pm1\}^E ε , ε ′ ∈ { ± 1 } E λ ⋅ Pr ( S E = s ) = 2 m Pr ( ∑ Y i = ( s , u ) ) \lambda\cdot\Pr(S_E=s)=2^m\Pr(\sum Y_i=(s,u)) λ ⋅ Pr ( S E = s ) = 2 m Pr ( ∑ Y i = ( s , u )) Y i = ( w i , ε i w i ) Y_i=(w_i,\varepsilon_iw_i) Y i = ( w i , ε i w i ) E [ Z 2 ∣ S E = s ] ⋅ Pr ( S E = s ) = 4 m Pr ( ∑ X i = ( s , u , u ) ) \mathbb{E}[Z^2\mid S_E=s]\cdot\Pr(S_E=s)=4^m\Pr(\sum X_i=(s,u,u)) E [ Z 2 ∣ S E = s ] ⋅ Pr ( S E = s ) = 4 m Pr ( ∑ X i = ( s , u , u )) X i = ( w i , ε i w i , ε i ′ w i ) X_i=(w_i,\varepsilon_iw_i,\varepsilon'_iw_i) X i = ( w i , ε i w i , ε i ′ w i ) m m m X X X ( μ , 0 , 0 ) (\mu,0,0) ( μ , 0 , 0 ) diag ( σ 2 , E w 2 , E w 2 ) \operatorname{diag}(\sigma^2,\mathbb{E}w^2,\mathbb{E}w^2) diag ( σ 2 , E w 2 , E w 2 ) { a ≡ b ≡ c ( m o d 2 ) } \{a\equiv b\equiv c\pmod2\} { a ≡ b ≡ c ( mod 2 )} Y Y Y { a ≡ b ( m o d 2 ) } \{a\equiv b\pmod2\} { a ≡ b ( mod 2 )} 1 + ∑ r = 1 3 m − r / 2 P r + O ( m − 2 ) 1+\sum_{r=1}^{3}m^{-r/2}P_r+O(m^{-2}) 1 + ∑ r = 1 3 m − r /2 P r + O ( m − 2 ) ( 4 ⋅ 1 / 2 2 ) (4\cdot1/2^2) ( 4 ⋅ 1/ 2 2 ) κ 122 = κ 133 = Cov ( w , w 2 ) / ( σ E w 2 ) \kappa_{122}=\kappa_{133}=\operatorname{Cov}(w,w^2)/(\sigma\mathbb{E}w^2) κ 122 = κ 133 = Cov ( w , w 2 ) / ( σ E w 2 ) κ 2222 = κ 3333 = E w 4 / ( E w 2 ) 2 − 3 \kappa_{2222}=\kappa_{3333}=\mathbb{E}w^4/(\mathbb{E}w^2)^2-3 κ 2222 = κ 3333 = E w 4 / ( E w 2 ) 2 − 3 κ 1122 = κ 1133 \kappa_{1122}=\kappa_{1133} κ 1122 = κ 1133 κ 2233 = E w 4 / ( E w 2 ) 2 − 1 \kappa_{2233}=\mathbb{E}w^4/(\mathbb{E}w^2)^2-1 κ 2233 = E w 4 / ( E w 2 ) 2 − 1 ε w \varepsilon w εw ε , ε ′ \varepsilon,\varepsilon' ε , ε ′
Order m − 1 / 2 m^{-1/2} m − 1/2 P 1 P_1 P 1 − κ 122 z -\kappa_{122}z − κ 122 z X X X − κ 122 z / 2 -\kappa_{122}z/2 − κ 122 z /2 Y Y Y S E S_E S E P 1 ( X ) + P 1 ( S E ) − 2 P 1 ( Y ) P_1(X)+P_1(S_E)-2P_1(Y) P 1 ( X ) + P 1 ( S E ) − 2 P 1 ( Y )
Order m − 1 m^{-1} m − 1 6 κ 2233 / 24 = κ 6\kappa_{2233}/24=\kappa 6 κ 2233 /24 = κ P 2 P_2 P 2 κ 122 2 ( z 2 − 1 ) / 4 \kappa_{122}^2(z^2-1)/4 κ 122 2 ( z 2 − 1 ) /4 − 1 2 ( P 1 ( X ) 2 + P 1 ( S E ) 2 − 2 P 1 ( Y ) 2 ) = − κ 122 2 z 2 / 4 -\frac{1}{2}(P_1(X)^2+P_1(S_E)^2-2P_1(Y)^2)=-\kappa_{122}^2z^2/4 − 2 1 ( P 1 ( X ) 2 + P 1 ( S E ) 2 − 2 P 1 ( Y ) 2 ) = − κ 122 2 z 2 /4 κ − κ 122 2 / 4 = κ c \kappa-\kappa_{122}^2/4=\kappa_c κ − κ 122 2 /4 = κ c z z z
Order m − 3 / 2 m^{-3/2} m − 3/2 z z z γ M \gamma_M γ M z z z ∑ w i 2 \sum w_i^2 ∑ w i 2 S E S_E S E ( σ B / Var ( w ) ) z / m (\sigma_B/\operatorname{Var}(w))z/\sqrt{m} ( σ B / Var ( w )) z / m m A mA m A ∑ w i 2 \sum w_i^2 ∑ w i 2
The unconditional statement is the same computation with X = ( ε w , ε ′ w ) X=(\varepsilon w,\varepsilon'w) X = ( εw , ε ′ w ) S E ≡ u ( m o d 2 ) S_E\equiv u\pmod2 S E ≡ u ( mod 2 ) Y = ε w Y=\varepsilon w Y = εw m − 1 m^{-1} m − 1 κ 1122 / 4 = κ \kappa_{1122}/4=\kappa κ 1122 /4 = κ c 2 ( M , u ) c_2(M,u) c 2 ( M , u ) m − 2 m^{-2} m − 2 u u u c 2 ( M , u ) = c 2 ( M , 0 ) − 2 κ u 2 / E w 2 c_2(M,u)=c_2(M,0)-2\kappa u^2/\mathbb{E}w^2 c 2 ( M , u ) = c 2 ( M , 0 ) − 2 κ u 2 / E w 2
Heuristically, the sub-block first moments are proportional to ( ∑ w i 2 ) − 1 / 2 (\sum w_i^2)^{-1/2} ( ∑ w i 2 ) − 1/2 ∑ w i 2 \sum w_i^2 ∑ w i 2 κ / m \kappa/m κ / m δ n 2 \delta_n^2 δ n 2 S E S_E S E ∑ w i 2 \sum w_i^2 ∑ w i 2 ∑ w i \sum w_i ∑ w i M M M 4 / 45 − 1 / 12 = 1 / 180 4/45-1/12=1/180 4/45 − 1/12 = 1/180 κ c = ( 1 / 180 ) / ( 4 / 9 ) = 1 / 80 \kappa_c=(1/180)/(4/9)=1/80 κ c = ( 1/180 ) / ( 4/9 ) = 1/80 m = 12 m=12 m = 12 M = 32 M=32 M = 32 10 6 10^6 1 0 6 + 0.013 +0.013 + 0.013 M = ∞ M=\infty M = ∞ − 0.003 ± 0.001 -0.003\pm0.001 − 0.003 ± 0.001 + 0.001 +0.001 + 0.001 m m m − 0.4 / m 2 -0.4/m^2 − 0.4/ m 2 S E S_E S E b_prime_exact.py): the unconditional remainder times m 2 m^2 m 2 M = 16 M=16 M = 16 m = 80 m=80 m = 80 m = 80 m=80 m = 80 m = 40 m=40 m = 40 c 2 = 0.246 c_2=0.246 c 2 = 0.246 c 2 ( M , 0 ) c_2(M,0) c 2 ( M , 0 ) M = 16 M=16 M = 16 m 2 m^2 m 2 − 0.008 -0.008 − 0.008 u = 8 u=8 u = 8 − 0.771 -0.771 − 0.771 u = 16 u=16 u = 16 − 0.009 -0.009 − 0.009 − 0.772 -0.772 − 0.772 M = 16 M=16 M = 16 m 2 m^2 m 2 m = 28 , 40 , 56 m=28,40,56 m = 28 , 40 , 56 z = 1 z=1 z = 1 z = 2 z=2 z = 2 z z z ⋅ m 3 / 2 / z = − 0.0104 , − 0.0128 , − 0.0138 , − 0.0145 \cdot m^{3/2}/z=-0.0104,-0.0128,-0.0138,-0.0145 ⋅ m 3/2 / z = − 0.0104 , − 0.0128 , − 0.0138 , − 0.0145 m = 20 , 40 , 80 , 120 m=20,40,80,120 m = 20 , 40 , 80 , 120 γ 16 = − 0.0171 \gamma_{16}=-0.0171 γ 16 = − 0.0171 m − 1 / 2 m^{-1/2} m − 1/2
Proposition B''' (the identity evaluated exactly). For moderate M M M S k S_k S k ∣ s − m μ ∣ ≤ 2 σ m |s-m\mu|\le2\sigma\sqrt{m} ∣ s − m μ ∣ ≤ 2 σ m ∣ u ∣ ≤ M |u|\le M ∣ u ∣ ≤ M
Table 11. For these M M M ≤ 1 2 \le\frac{1}{2} ≤ 2 1 λ ≥ 5 \lambda\ge5 λ ≥ 5 2 / λ 2/\lambda 2/ λ κ c / m \kappa_c/m κ c / m
The theorems
Theorem R (random weights). Let w 1 , … , w n w_1,\ldots,w_n w 1 , … , w n { 1 , … , M } \{1,\ldots,M\} { 1 , … , M } M = 2 b M=2^b M = 2 b g ≥ 0 g\ge0 g ≥ 0 T T T H H H S E S_E S E m m m λ m ≥ λ ∗ \lambda_m\ge\lambda^* λ m ≥ λ ∗ ∣ T − S / 2 ∣ ≤ S H / 2 \lvert T-S/2\rvert\le S_H/2 ∣ T − S /2 ∣ ≤ S H /2 H H H E E E 2 / λ ∗ + O ( m − 1 / 5 ) + 2 e − g 2 / 6 2/\lambda^*+O(m^{-1/5})+2e^{-g^2/6} 2/ λ ∗ + O ( m − 1/5 ) + 2 e − g 2 /6 M ≥ 2 M\ge2 M ≥ 2 O ( n log n ) + O ( λ ∗ M ⋅ m 5 / 4 ) O(n\log n)+O(\sqrt{\lambda^*M}\mathbin{\cdot}m^{5/4}) O ( n log n ) + O ( λ ∗ M ⋅ m 5/4 )
Proof. Lemma A places the residual within M / 2 M/2 M /2 S E / 2 S_E/2 S E /2 T T T E E E S E S_E S E u u u S E S_E S E g g g ≥ 1 − 2 e − g 2 / 6 \ge1-2e^{-g^2/6} ≥ 1 − 2 e − g 2 /6 H H H E E E S E S_E S E λ ≥ λ ∗ ( 1 − O ( m − 1 / 5 ) ) \lambda\ge\lambda^*(1-O(m^{-1/5})) λ ≥ λ ∗ ( 1 − O ( m − 1/5 )) 2 / λ ∗ + O ( m − 1 / 5 ) 2/\lambda^*+O(m^{-1/5}) 2/ λ ∗ + O ( m − 1/5 ) T T T
Theorem R′ (failure probability δ \delta δ Fix r = ⌈ log 2 ( 2 / δ ) ⌉ r=\lceil\log_2(2/\delta)\rceil r = ⌈ log 2 ( 2/ δ )⌉ E 1 , … , E r E_1,\ldots,E_r E 1 , … , E r G G G ∣ T − S / 2 ∣ ≤ S G / 2 − r m M \lvert T-S/2\rvert\le S_G/2-rmM ∣ T − S /2 ∣ ≤ S G /2 − r m M T T T G G G j j j T j T_j T j G ∪ E j G\cup E_j G ∪ E j ∣ T j − ( S G + S E j ) / 2 ∣ ≤ S G / 2 + M / 2 \lvert T_j-(S_G+S_{E_j})/2\rvert\le S_G/2+M/2 ∣ T j − ( S G + S E j ) /2 ∣ ≤ S G /2 + M /2 G G G ≤ M \le M ≤ M E j E_j E j g = 6 log ( 4 r / δ ) g=\sqrt{6\log(4r/\delta)} g = 6 log ( 4 r / δ ) λ m ≥ 8 \lambda_m\ge8 λ m ≥ 8 m m m ≤ 1 2 \le\frac{1}{2} ≤ 2 1 ∣ T − S / 2 ∣ ≤ S G / 2 − r m M \lvert T-S/2\rvert\le S_G/2-rmM ∣ T − S /2 ∣ ≤ S G /2 − r m M ≤ δ \le\delta ≤ δ O ( log ( 1 / δ ) ⋅ ( n log n + λ m M m 5 / 4 ) ) O(\log(1/\delta)\mathbin{\cdot}(n\log n+\sqrt{\lambda_mM}m^{5/4})) O ( log ( 1/ δ ) ⋅ ( n log n + λ m M m 5/4 ))
Proof. By Lemma A with head G G G j j j ∣ u j ∣ ≤ M \lvert u_j\rvert\le M ∣ u j ∣ ≤ M σ \sigma σ G G G ≤ 1 2 \le\frac{1}{2} ≤ 2 1 λ ≥ 6 \lambda\ge6 λ ≥ 6 λ m ≥ 8 \lambda_m\ge8 λ m ≥ 8 ≤ 2 r e − g 2 / 6 = δ / 2 \le2re^{-g^2/6}=\delta/2 ≤ 2 r e − g 2 /6 = δ /2
For b = O ( log n ) b=O(\log n) b = O ( log n ) ( log n ) 2 ≪ b ≪ n (\log n)^2\ll b\ll n ( log n ) 2 ≪ b ≪ n 2 b / 2 + o ( b ) 2^{b/2+o(b)} 2 b /2 + o ( b )
Dense instances: the arcs and the almost divisors
Lemma C. Let d ≥ 2 d \ge2 d ≥ 2 r ( c ) = # { x : ∑ i x i w i ≡ c ( m o d d ) } r(c)=\#\{x:\sum_i x_iw_i \equiv c \pmod d\} r ( c ) = # { x : ∑ i x i w i ≡ c ( mod d )} d ′ ∣ d d' \mid d d ′ ∣ d k ( d ′ ) = # { i : d ′ ∤ w i } k(d')=\#\{i:d'\nmid w_i\} k ( d ′ ) = # { i : d ′ ∤ w i }
∣ d ⋅ r ( c ) / 2 n − 1 ∣ ≤ ∑ a = 1 d − 1 ∏ i ∣ cos ( π a w i / d ) ∣ ≤ ∑ d ′ ∣ d , d ′ > 1 φ ( d ′ ) e − π 2 k ( d ′ ) / ( 2 d ′ 2 ) \left|d\cdot r(c)/2^n-1\right|\le\sum_{a=1}^{d-1}\prod_i\left|\cos(\pi aw_i/d)\right|\le\sum_{d'\mid d,\ d'>1}\varphi(d')e^{-\pi^2k(d')/(2d'^2)} ∣ d ⋅ r ( c ) / 2 n − 1 ∣ ≤ ∑ a = 1 d − 1 ∏ i ∣ cos ( π a w i / d ) ∣ ≤ ∑ d ′ ∣ d , d ′ > 1 φ ( d ′ ) e − π 2 k ( d ′ ) / ( 2 d ′2 )
If all but k k k d d d 2 k < d 2^k<d 2 k < d d − 2 k d-2^k d − 2 k a / d a/d a / d a ≠ 0 a\ne0 a = 0 ∏ i ∣ cos ( π a w i / d ) ∣ ≥ 1 / ( d − 1 ) \prod_i|\cos(\pi aw_i/d)|\ge1/(d-1) ∏ i ∣ cos ( π a w i / d ) ∣ ≥ 1/ ( d − 1 )
Proof. (1) r ( c ) = d − 1 ∑ a e ( − a c / d ) ∏ i ( 1 + e ( a w i / d ) ) r(c)=d^{-1}\sum_a e(-ac/d)\prod_i(1+e(aw_i/d)) r ( c ) = d − 1 ∑ a e ( − a c / d ) ∏ i ( 1 + e ( a w i / d )) a / d = a ′ / d ′ a/d=a'/d' a / d = a ′ / d ′ d ′ ∤ w i d'\nmid w_i d ′ ∤ w i ≤ cos ( π / d ′ ) ≤ e − π 2 / ( 2 d ′ 2 ) \le\cos(\pi/d')\le e^{-\pi^2/(2d'^2)} ≤ cos ( π / d ′ ) ≤ e − π 2 / ( 2 d ′2 ) a ≠ 0 a\ne0 a = 0 − 1 -1 − 1
Corollary C′ (via Bringmann–Wellnitz). Let X X X n n n Σ X \Sigma_X Σ X μ X \mu_X μ X X X X C δ C_\delta C δ C δ = 1699200 log ( 2 n ) log 2 ( 2 μ X ) C_\delta=1699200\log(2n)\log^2(2\mu_X) C δ = 1699200 log ( 2 n ) log 2 ( 2 μ X ) ( 4 + 2 C g ) μ X max X Σ X / n 2 ≤ t ≤ Σ X / 2 (4+2C_g)\mu_X\max X\Sigma_X/n^2\le t\le\Sigma_X/2 ( 4 + 2 C g ) μ X max X Σ X / n 2 ≤ t ≤ Σ X /2 C g = 169920 log ( 2 μ X ) C_g=169920\log(2\mu_X) C g = 169920 log ( 2 μ X ) d ′ ≤ D = ⌊ 4 μ X Σ X / n 2 ⌋ d'\le D=\lfloor4\mu_X\Sigma_X/n^2\rfloor d ′ ≤ D = ⌊ 4 μ X Σ X / n 2 ⌋ r d ′ ( c ) r_{d'}(c) r d ′ ( c ) a / d ′ a/d' a / d ′ t t t r d ′ ( t m o d d ′ ) > 0 r_{d'}(t\bmod d')>0 r d ′ ( t mod d ′ ) > 0 d ′ ≤ D d'\le D d ′ ≤ D
Proof. Necessity is clear. By [BW, Theorems 4.1 and 4.3] there is d ≤ 4 μ X Σ X / n 2 d\le4\mu_X\Sigma_X/n^2 d ≤ 4 μ X Σ X / n 2 t ∈ S X t\in\mathcal{S}_X t ∈ S X t m o d d ∈ S X m o d d t\bmod d\in\mathcal{S}_X\bmod d t mod d ∈ S X mod d t > Σ X / 2 t>\Sigma_X/2 t > Σ X /2
The decider costs O ( n D 2 ) = O ( n 3 ) O(nD^2)=O(n^3) O ( n D 2 ) = O ( n 3 ) O ~ ( n ) \widetilde{O}(n) O ( n ) α μ X Σ X / n 2 \alpha\mu_X\Sigma_X/n^2 α μ X Σ X / n 2 log 2 d \log_2d log 2 d
Corollary C′′ (constructor on dense instances). Under the hypotheses of Corollary C′ a solution is constructed in O ( n ⋅ Σ X ) = O ( n 4 / ( C δ μ X ) ) O(n\cdot\Sigma_X)=O(n^4/(C_\delta\mu_X)) O ( n ⋅ Σ X ) = O ( n 4 / ( C δ μ X ))
Proof. C δ C_\delta C δ n 2 ≥ C δ μ X max X n^2\ge C_\delta\mu_X\max X n 2 ≥ C δ μ X max X Σ X ≤ n ⋅ max X ≤ n 3 / ( C δ μ X ) \Sigma_X\le n\cdot\max X\le n^3/(C_\delta\mu_X) Σ X ≤ n ⋅ max X ≤ n 3 / ( C δ μ X ) [ 0 , Σ X ] [0,\Sigma_X] [ 0 , Σ X ] t t t
The constructor does not use the arcs: on dense instances the sums are polynomially bounded. A constructor through the class counts alone, by self-reduction, would need the hypotheses of [BW] to persist along the descent.
Las Vegas on random instances
The algorithm already has one-sided failure: outputs are verified, failure is reported, and Theorem R′ makes it ≤ δ \le\delta ≤ δ × log ( 1 / δ ) \times\log(1/\delta) × log ( 1/ δ ) r r r p ≤ 2 − m / 2 p\le2^{-m/2} p ≤ 2 − m /2 M = 64 M=64 M = 64 S E S_E S E Pr ( Z = 0 ) = 0.0075 , 0.0011 , 1.2 ⋅ 10 − 4 \Pr(Z=0)=0.0075,0.0011,1.2\cdot10^{-4} Pr ( Z = 0 ) = 0.0075 , 0.0011 , 1.2 ⋅ 1 0 − 4 λ = 6.7 , 12.7 , 24.4 \lambda=6.7,12.7,24.4 λ = 6.7 , 12.7 , 24.4 λ − 3 \lambda^{-3} λ − 3 E ( Z / λ − 1 ) 4 ≈ ( 3.5 − 6.5 ) / λ 2 E(Z/\lambda-1)^4\approx(3.5-6.5)/\lambda^2 E ( Z / λ − 1 ) 4 ≈ ( 3.5 − 6.5 ) / λ 2 m ≥ ( 4 / 3 ) ( b + O ( log m ) ) m\ge(4/3)(b+O(\log m)) m ≥ ( 4/3 ) ( b + O ( log m )) e − λ e^{-\lambda} e − λ 3 ⋅ 10 − 6 3\cdot10^{-6} 3 ⋅ 1 0 − 6 λ = 12.7 \lambda=12.7 λ = 12.7 λ \lambda λ κ c / m \kappa_c/m κ c / m
Theorem LV (exponential tail of the endgame count). In the setting of Lemma B, with M = 2 b M = 2^b M = 2 b ∣ s − m μ ∣ ≤ g σ m |s-m\mu| \le g\sigma\sqrt{m} ∣ s − m μ ∣ ≤ g σ m ∣ u ∣ ≤ M |u| \le M ∣ u ∣ ≤ M u ≡ s ( m o d 2 ) u \equiv s \pmod{2} u ≡ s ( mod 2 ) m 0 m_0 m 0 C g C_g C g m ≥ m 0 m \ge m_0 m ≥ m 0
Pr ( Z ( u ) = 0 ∣ S E = s ) ≤ C g ( m 5 / 2 M 5 ρ m + 2 − 0.53 m ) , ρ = sup M ≥ 2 sup ∥ θ ∥ ≥ 1 2 M , ∥ θ − 1 2 ∥ ≥ 1 4 M E cos 4 ( π w θ ) ≤ 0.446. \Pr(Z(u)=0 \mid S_E=s) \le C_g\left(m^{5/2}M^5\rho^m+2^{-0.53m}\right), \qquad\rho=\sup_{M\ge2}\ \sup_{\lVert\theta\rVert\ge\frac{1}{2M},\ \lVert\theta-\frac{1}{2}\rVert\ge\frac{1}{4M}} \mathbb{E}\cos^4(\pi w\theta)\le0.446. Pr ( Z ( u ) = 0 ∣ S E = s ) ≤ C g ( m 5/2 M 5 ρ m + 2 − 0.53 m ) , ρ = M ≥ 2 sup ∥ θ ∥ ≥ 2 M 1 , ∥ θ − 2 1 ∥ ≥ 4 M 1 sup E cos 4 ( π w θ ) ≤ 0.446. The bound ρ ≤ 0.44495 \rho\le0.44495 ρ ≤ 0.44495 M ≥ 2 M \ge2 M ≥ 2 lv_rate_certified.py); it enters the proof only through the last item.
Hence the per-round failure is at most 2 − m / 2 2^{-m/2} 2 − m /2 m ≥ 7.6 ( b + O ( log m ) ) m \ge7.6(b+O(\log m)) m ≥ 7.6 ( b + O ( log m ))
Proof. Put a = ( s + u ) / 2 a=(s+u)/2 a = ( s + u ) /2 V = ∑ E w i 2 V=\sum_{E}w_i^2 V = ∑ E w i 2
Z ( u ) = ∫ 0 1 F ( θ ) e ( − a θ ) d θ = 2 m ∫ − 1 / 2 1 / 2 cos ( π u θ ) P ( θ ) d θ , P ( θ ) = ∏ i ∈ E cos ( π w i θ ) . Z(u)=\int_0^1 F(\theta)e(-a\theta)\,d\theta=2^m\int_{-1/2}^{1/2}\cos(\pi u\theta)P(\theta)\,d\theta,\qquad P(\theta)=\prod_{i\in E}\cos(\pi w_i\theta). Z ( u ) = ∫ 0 1 F ( θ ) e ( − a θ ) d θ = 2 m ∫ − 1/2 1/2 cos ( π u θ ) P ( θ ) d θ , P ( θ ) = i ∈ E ∏ cos ( π w i θ ) . which holds because s ≡ u ( m o d 2 ) s\equiv u\pmod{2} s ≡ u ( mod 2 ) [ 0 , 1 2 ] [0,\frac{1}{2}] [ 0 , 2 1 ]
the major arc [ 0 , 1 / ( 2 M ) ] [0,1/(2M)] [ 0 , 1/ ( 2 M )]
the rest B = [ 1 / ( 2 M ) , 1 2 − 1 / ( 4 M ) ] \mathcal{B}=[1/(2M),\frac{1}{2}-1/(4M)] B = [ 1/ ( 2 M ) , 2 1 − 1/ ( 4 M )]
the parity window [ 1 2 − 1 / ( 4 M ) , 1 2 ] [\frac{1}{2}-1/(4M),\frac{1}{2}] [ 2 1 − 1/ ( 4 M ) , 2 1 ]
Put
λ 0 ( w ) = 2 m ( 2 π V ) 1 / 2 , λ min = 2 m ( 2 π m M 2 ) 1 / 2 . \lambda_0(w)=2^m\left(\frac{2}{\pi V}\right)^{1/2},\qquad\lambda_{\min}=2^m\left(\frac{2}{\pi mM^2}\right)^{1/2}. λ 0 ( w ) = 2 m ( π V 2 ) 1/2 , λ m i n = 2 m ( π m M 2 2 ) 1/2 . Since V ≤ m M 2 V\le mM^2 V ≤ m M 2 λ 0 ( w ) ≥ λ min \lambda_0(w)\ge\lambda_{\min} λ 0 ( w ) ≥ λ m i n
Good weights. Let k 1 = # { i : w i > M / 2 } k_1=\#\{i:w_i>M/2\} k 1 = # { i : w i > M /2 } j = # { i : w i odd } j=\#\{i:w_i\text{ odd}\} j = # { i : w i odd } G = { k 1 ≥ m / 20 , j ≥ m / 20 } G=\{k_1\ge m/20,j\ge m/20\} G = { k 1 ≥ m /20 , j ≥ m /20 } G G G V ≥ k 1 M 2 / 4 ≥ m M 2 / 80 V\ge k_1M^2/4\ge mM^2/80 V ≥ k 1 M 2 /4 ≥ m M 2 /80
Case M ≥ 4 M\ge4 M ≥ 4 Write M = 2 M ′ M=2M' M = 2 M ′ w i = M ′ b i + y i w_i=M'b_i+y_i w i = M ′ b i + y i b i = 1 [ w i > M / 2 ] b_i=1[w_i>M/2] b i = 1 [ w i > M /2 ] y i y_i y i { 1 , … , M ′ } \{1,\ldots,M'\} { 1 , … , M ′ } b i b_i b i ( b i ) (b_i) ( b i ) S E = M ′ k 1 + ∑ i y i S_E=M'k_1+\sum_i y_i S E = M ′ k 1 + ∑ i y i m m m M ′ ≥ 2 M'\ge2 M ′ ≥ 2 13 / ( σ ′ m ) 13/(\sigma'\sqrt{m}) 13/ ( σ ′ m ) σ ′ 2 = ( M ′ 2 − 1 ) / 12 {\sigma'}^2=(M'^2-1)/12 σ ′ 2 = ( M ′2 − 1 ) /12
Pr ( k 1 < m / 20 , S E = s ) ≤ 2 − m # { b ∈ { 0 , 1 } m : 20 ∣ b ∣ < m } ⋅ 13 σ ′ m . \Pr(k_1<m/20,S_E=s)\le2^{-m}\#\{b\in\{0,1\}^m:20|b|<m\}\cdot\frac{13}{\sigma'\sqrt{m}}. Pr ( k 1 < m /20 , S E = s ) ≤ 2 − m # { b ∈ { 0 , 1 } m : 20∣ b ∣ < m } ⋅ σ ′ m 13 . The same holds for j j j w i = 2 y i − 1 + p i w_i=2y_i-1+p_i w i = 2 y i − 1 + p i p i ∈ { 0 , 1 } p_i\in\{0,1\} p i ∈ { 0 , 1 } y i y_i y i { 1 , … , M ′ } \{1,\ldots,M'\} { 1 , … , M ′ } ( p i ) (p_i) ( p i ) S E = 2 ∑ i y i − # { i : p i = 0 } S_E=2\sum_i y_i-\#\{i:p_i=0\} S E = 2 ∑ i y i − # { i : p i = 0 }
# { b ∈ { 0 , 1 } m : 20 ∣ b ∣ < m } ≤ ∑ b x m − 20 ∣ b ∣ = ( x ( 1 + x − 20 ) ) m ≤ ( 5 / 4 ) m \#\{b \in\{0,1\}^{m}: 20|b| < m\} \le\sum_b x^{m-20|b|} = \left(x(1+x^{-20})\right)^m \le(5/4)^m # { b ∈ { 0 , 1 } m : 20∣ b ∣ < m } ≤ b ∑ x m − 20∣ b ∣ = ( x ( 1 + x − 20 ) ) m ≤ ( 5/4 ) m So each of the two events has probability at most ( 5 / 8 ) m ⋅ 13 / ( σ ′ m ) (5/8)^m \cdot13/(\sigma'\sqrt{m}) ( 5/8 ) m ⋅ 13/ ( σ ′ m )
Dividing by Pr ( S E = s ) \Pr(S_E=s) Pr ( S E = s ) Pr ( S E = s ) ≥ e − g 2 / 2 / ( 2 2 π σ m ) \Pr(S_E=s) \ge e^{-g^2/2}/(2\sqrt{2\pi}\sigma\sqrt{m}) Pr ( S E = s ) ≥ e − g 2 /2 / ( 2 2 π σ m ) 17 2 π e g 2 / 2 / m ≤ 1 / 2 17\sqrt{2\pi}e^{g^2/2}/m \le1/2 17 2 π e g 2 /2 / m ≤ 1/2 σ ≤ 3 σ ′ \sigma\le3\sigma' σ ≤ 3 σ ′
Pr ( G c ∣ S E = s ) ≤ 156 2 π e g 2 / 2 ( 5 / 8 ) m ≤ 156 2 π e g 2 / 2 2 − 0.53 m , \Pr(G^{\mathrm{c}}\mid S_E=s) \le156\sqrt{2\pi}e^{g^2/2}(5/8)^m \le156\sqrt{2\pi}e^{g^2/2}2^{-0.53m}, Pr ( G c ∣ S E = s ) ≤ 156 2 π e g 2 /2 ( 5/8 ) m ≤ 156 2 π e g 2 /2 2 − 0.53 m , because 5 / 8 ≤ 2 − 0.53 5/8 \le2^{-0.53} 5/8 ≤ 2 − 0.53
Case M = 2 M=2 M = 2 Each w i w_i w i S E + j = 2 m S_E+j=2m S E + j = 2 m s ≤ 3 m / 2 + g m / 2 s \le3m/2+g\sqrt{m}/2 s ≤ 3 m /2 + g m /2 j ≥ m / 4 ≥ m / 20 j \ge m/4 \ge m/20 j ≥ m /4 ≥ m /20 m ≥ 4 g 2 m \ge4g^2 m ≥ 4 g 2 k 1 = m − j = s − m ≥ m / 2 − g m / 2 ≥ m / 4 k_1=m-j=s-m \ge m/2-g\sqrt{m}/2 \ge m/4 k 1 = m − j = s − m ≥ m /2 − g m /2 ≥ m /4 V ≥ m = m M 2 / 4 V \ge m=mM^2/4 V ≥ m = m M 2 /4 G G G S E = s S_E=s S E = s Pr ( G c ∣ S E = s ) = 0 \Pr(G^{\mathrm{c}}\mid S_E=s)=0 Pr ( G c ∣ S E = s ) = 0
∣ g ( θ ) − P ( θ ) ∣ ≤ π 4 θ 4 4 ∑ i w i 4 ⋅ g ( θ ) . |g(\theta)-P(\theta)|\le\frac{\pi^4\theta^4}{4}\sum_i w_i^4\cdot g(\theta). ∣ g ( θ ) − P ( θ ) ∣ ≤ 4 π 4 θ 4 i ∑ w i 4 ⋅ g ( θ ) . On 1 / ( π M ) ≤ θ ≤ 1 / ( 2 M ) 1/(\pi M)\le\theta\le1/(2M) 1/ ( π M ) ≤ θ ≤ 1/ ( 2 M ) ∣ x i ∣ ≤ π / 2 |x_i|\le\pi/2 ∣ x i ∣ ≤ π /2 0 ≤ P ≤ g 0\le P\le g 0 ≤ P ≤ g cos x ≤ e − x 2 / 2 \cos x\le e^{-x^2/2} cos x ≤ e − x 2 /2 ∣ x ∣ ≤ 2 |x|\le2 ∣ x ∣ ≤ 2 ∣ x ∣ ≤ 1 |x|\le1 ∣ x ∣ ≤ 1 cos x = 2 cos 2 ( x / 2 ) − 1 \cos x=2\cos^2(x/2)-1 cos x = 2 cos 2 ( x /2 ) − 1
We compare with Gaussian integrals rather than with λ ( w ) \lambda(w) λ ( w ) cos ( π u θ ) ≥ 1 − π 2 u 2 θ 2 / 2 \cos(\pi u\theta)\ge1-\pi^2u^2\theta^2/2 cos ( π u θ ) ≥ 1 − π 2 u 2 θ 2 /2 ∫ 0 ∞ θ 2 g = ( 2 π 2 V ) − 1 ∫ 0 ∞ g \int_0^\infty\theta^2g=(2\pi^2V)^{-1}\int_0^\infty g ∫ 0 ∞ θ 2 g = ( 2 π 2 V ) − 1 ∫ 0 ∞ g ∫ 0 ∞ θ 4 g = 3 ( π 2 V ) − 2 ∫ 0 ∞ g \int_0^\infty\theta^4g=3(\pi^2V)^{-2}\int_0^\infty g ∫ 0 ∞ θ 4 g = 3 ( π 2 V ) − 2 ∫ 0 ∞ g ∫ θ 0 ∞ g ≤ g ( θ 0 ) / ( π 2 V θ 0 ) \int_{\theta_0}^\infty g\le g(\theta_0)/(\pi^2V\theta_0) ∫ θ 0 ∞ g ≤ g ( θ 0 ) / ( π 2 V θ 0 ) θ 0 = 1 / ( π M ) \theta_0=1/(\pi M) θ 0 = 1/ ( π M )
2 m + 1 ∫ 0 1 / ( 2 M ) cos ( π u θ ) P ( θ ) d θ ≥ λ 0 ( w ) ( 1 − u 2 2 V − 3 ∑ i w i 4 4 V 2 − η 3 ) , η 3 = 3 2 e − V / ( 2 M 2 ) π V / M 2 2^{m+1}\int_0^{1/(2M)}\cos(\pi u\theta)P(\theta)\,d\theta
\ge\lambda_0(w)\left(1-\frac{u^2}{2V}-\frac{3\sum_i w_i^4}{4V^2}-\eta_3\right),\qquad
\eta_3=\frac{3\sqrt{2}e^{-V/(2M^2)}}{\sqrt{\pi V/M^2}} 2 m + 1 ∫ 0 1/ ( 2 M ) cos ( π u θ ) P ( θ ) d θ ≥ λ 0 ( w ) ( 1 − 2 V u 2 − 4 V 2 3 ∑ i w i 4 − η 3 ) , η 3 = π V / M 2 3 2 e − V / ( 2 M 2 ) On G G G u 2 ≤ M 2 u^2\le M^2 u 2 ≤ M 2 V ≥ m M 2 / 80 V\ge mM^2/80 V ≥ m M 2 /80 u 2 / ( 2 V ) ≤ 40 / m u^2/(2V)\le40/m u 2 / ( 2 V ) ≤ 40/ m ∑ w i 4 ≤ M 2 V \sum w_i^4\le M^2V ∑ w i 4 ≤ M 2 V 3 ∑ w i 4 / ( 4 V 2 ) ≤ 60 / m 3\sum w_i^4/(4V^2)\le60/m 3 ∑ w i 4 / ( 4 V 2 ) ≤ 60/ m V / M 2 ≥ m / 80 V/M^2\ge m/80 V / M 2 ≥ m /80 e − t / 2 ≤ 2 / t e^{-t/2}\le2/t e − t /2 ≤ 2/ t η 3 ≤ 6 2 ⋅ M 2 / V ≤ 480 2 / m \eta_3\le6\sqrt{2}\cdot M^2/V\le480\sqrt{2}/m η 3 ≤ 6 2 ⋅ M 2 / V ≤ 480 2 / m
For m ≥ 12000 m\ge12000 m ≥ 12000 ≥ 3 / 4 \ge3/4 ≥ 3/4 ( 3 / 4 ) λ 0 ( w ) ≥ ( 3 / 4 ) λ min (3/4)\lambda_0(w)\ge(3/4)\lambda_{\min} ( 3/4 ) λ 0 ( w ) ≥ ( 3/4 ) λ m i n
Parity window, on $G. For w w w ∣ cos ( π w ( 1 / 2 − δ ) ) ∣ ≤ 1 |\cos(\pi w(1/2-\delta))|\le1 ∣ cos ( π w ( 1/2 − δ )) ∣ ≤ 1 w w w ∣ sin ( π w δ ) ∣ ≤ sin ( π / 4 ) |\sin(\pi w\delta)|\le\sin(\pi/4) ∣ sin ( π w δ ) ∣ ≤ sin ( π /4 ) ∣ P ∣ ≤ 2 − j / 2 ≤ 2 − m / 40 |P|\le2^{-j/2}\le2^{-m/40} ∣ P ∣ ≤ 2 − j /2 ≤ 2 − m /40 1 / ( 2 M ) 1/(2M) 1/ ( 2 M )
2 m 2 − j / 2 / ( 2 M ) ≤ λ min / 4. 2^{m}2^{-j/2}/(2M) \le\lambda_{\min}/4. 2 m 2 − j /2 / ( 2 M ) ≤ λ m i n /4. This holds as soon as 2 j ≥ 2 π m 2^j \ge2\pi m 2 j ≥ 2 π m m ≤ 20 j m \le20j m ≤ 20 j 40 π j ≤ 2 j 40\pi j \le2^j 40 π j ≤ 2 j j ≥ 12 j \ge12 j ≥ 12
Pr ( 2 m + 1 R ( w ) ≥ λ min / 4 ) ≤ ρ m / 16 ( λ min / 2 m + 3 ) 4 = 64 π 2 m 2 M 4 ρ m . \Pr\left(2^{m+1}R(w) \ge\lambda_{\min}/4\right) \le\frac{\rho^m/16}{(\lambda_{\min}/2^{m+3})^4}=64\pi^2m^2M^4\rho^m. Pr ( 2 m + 1 R ( w ) ≥ λ m i n /4 ) ≤ ( λ m i n / 2 m + 3 ) 4 ρ m /16 = 64 π 2 m 2 M 4 ρ m . Dividing by Pr ( S E = s ) ≥ e − g 2 / 2 / ( 2 2 π σ m ) \Pr(S_E=s)\ge e^{-g^2/2}/(2\sqrt{2\pi}\sigma\sqrt{m}) Pr ( S E = s ) ≥ e − g 2 /2 / ( 2 2 π σ m ) σ ≤ M \sigma\le M σ ≤ M 128 π 2 2 π e g 2 / 2 m 5 / 2 M 5 ρ m 128\pi^2\sqrt{2\pi}e^{g^2/2}m^{5/2}M^5\rho^m 128 π 2 2 π e g 2 /2 m 5/2 M 5 ρ m
On G G G
Z ( u ) ≥ 3 4 λ 0 ( w ) − 1 4 λ min − 2 m + 1 R ( w ) > ( 3 4 − 1 4 − 1 4 ) λ min > 0. Z(u) \ge\frac{3}{4}\lambda_0(w)-\frac{1}{4}\lambda_{\min}-2^{m+1}R(w) > \left(\frac{3}{4}-\frac{1}{4}-\frac{1}{4}\right)\lambda_{\min}>0. Z ( u ) ≥ 4 3 λ 0 ( w ) − 4 1 λ m i n − 2 m + 1 R ( w ) > ( 4 3 − 4 1 − 4 1 ) λ m i n > 0. Hence, for m ≥ max ( 12000 , 4 g 2 , 34 2 π e g 2 / 2 ) m\ge\max(12000,4g^2,34\sqrt{2\pi}e^{g^2/2}) m ≥ max ( 12000 , 4 g 2 , 34 2 π e g 2 /2 )
Pr ( Z ( u ) = 0 ∣ S E = s ) ≤ 2 2 π e g 2 / 2 ( 78 + 64 π 2 ) ( m 5 / 2 M 5 ρ m + 2 − 0.53 m ) . \Pr(Z(u)=0\mid S_E=s)\le2\sqrt{2\pi}e^{g^2/2}(78+64\pi^2)\left(m^{5/2}M^5\rho^m+2^{-0.53m}\right). Pr ( Z ( u ) = 0 ∣ S E = s ) ≤ 2 2 π e g 2 /2 ( 78 + 64 π 2 ) ( m 5/2 M 5 ρ m + 2 − 0.53 m ) . The rate follows from 0.44619 ≤ 2 − 22 0.44619\le2^{-22} 0.44619 ≤ 2 − 22 2 − m / 2 2^{-m/2} 2 − m /2 m ≥ 7.6 ( b + log 2 m ) m\ge7.6(b+\log_2m) m ≥ 7.6 ( b + log 2 m ) m ≥ m 0 m\ge m_0 m ≥ m 0
The constant 7.6 7.6 7.6 4 / 3 4/3 4/3 λ − 3 \lambda^{-3} λ − 3 M 5 M^5 M 5 ( 1 − o ( 1 ) ) / ( 6 ln m M ) (1-o(1))/(6\ln mM) ( 1 − o ( 1 )) / ( 6 ln m M )
The filtered endgame: Theorem R″
Setting.
It is that of Lemma B: a block E E E m m m { 1 , … , M } \{1,\ldots,M\} { 1 , … , M }
A search scheme is a family P \mathcal{P} P ( y , z ) (y,z) ( y , z ) E E E § 7.6 uses P 1 = { ( x ∩ A , x ∩ B ) : x ⊆ E } \mathcal{P}_1=\{(x\cap A,x\cap B):x\subseteq E\} P 1 = {( x ∩ A , x ∩ B ) : x ⊆ E } A A A B B B ∣ A ∣ = h = ⌊ m / 2 ⌋ |A|=h=\lfloor m/2\rfloor ∣ A ∣ = h = ⌊ m /2 ⌋
Put
N = # { ( y , z ) ∈ P : S y + S z = a } , Z r = # { ( y , z ) ∈ P : S y + S z = a , S y = r ( m o d P ) } , Y P = ∑ r Z r 2 . N=\#\{(y,z)\in\mathcal{P}:S_y+S_z=a\},\qquad Z_r=\#\{(y,z)\in\mathcal{P}:S_y+S_z=a,\ S_y=r\pmod{P}\},\qquad Y_P=\sum_r Z_r^2. N = # {( y , z ) ∈ P : S y + S z = a } , Z r = # {( y , z ) ∈ P : S y + S z = a , S y = r ( mod P )} , Y P = r ∑ Z r 2 . For P 1 \mathcal{P}_1 P 1 N = Z ( u ) N=Z(u) N = Z ( u ) r r r Z / P \mathbb{Z}/P Z / P P P P Z r > 0 Z_r>0 Z r > 0
Lemma J (averaging over the residues). For every finitely supported law of the weights and of P P P
Pr ( round succeeds ) ≥ ( E N ) 2 E [ P ⋅ Y P ] . \Pr(\text{round succeeds})\ge\frac{(\mathbb{E}N)^2}{\mathbb{E}[P\cdot Y_P]}. Pr ( round succeeds ) ≥ E [ P ⋅ Y P ] ( E N ) 2 . Proof.
By Cauchy–Schwarz, N = ∑ r ∈ R P Z r ≤ ∣ R P ∣ 1 / 2 Y P 1 / 2 N=\sum_{r\in R_P}Z_r\le|R_P|^{1/2}Y_P^{1/2} N = ∑ r ∈ R P Z r ≤ ∣ R P ∣ 1/2 Y P 1/2 ∣ R P ∣ / P ≥ N 2 / ( P ⋅ Y P ) |R_P|/P\ge N^2/(P\cdot Y_P) ∣ R P ∣/ P ≥ N 2 / ( P ⋅ Y P ) 0 / 0 = 0 0/0=0 0/0 = 0 N = 0 N=0 N = 0 Y P = 0 Y_P=0 Y P = 0
The function ( z , t ) ↦ z 2 / t (z,t)\mapsto z^2/t ( z , t ) ↦ z 2 / t t > 0 t>0 t > 0 0 0 0 ( 0 , 0 ) (0,0) ( 0 , 0 ) + ∞ +\infty + ∞ t = 0 t=0 t = 0 z 2 z^2 z 2
The lemma uses nothing about the scheme or the weights.
Lemma K (a random prime). Let P P P [ P 0 , 2 P 0 ) [P_0,2P_0) [ P 0 , 2 P 0 ) P 0 ≥ 100 P_0\ge100 P 0 ≥ 100 P P P Δ ≠ 0 \Delta\ne0 Δ = 0 ∣ Δ ∣ ≤ X |\Delta|\le X ∣Δ∣ ≤ X
E [ P ⋅ 1 [ P ∣ Δ ] ] ≤ 6 ln X . \mathbb{E}[P\cdot1[P\mid\Delta]]\le6\ln X. E [ P ⋅ 1 [ P ∣ Δ ]] ≤ 6 ln X . Proof.
From π ( y ) > y / ln y \pi(y)>y/\ln y π ( y ) > y / ln y y ≥ 17 y\ge17 y ≥ 17 π ( y ) < 1.25506 ⋅ y / ln y \pi(y)<1.25506\cdot y/\ln y π ( y ) < 1.25506 ⋅ y / ln y π ( 2 P 0 ) − π ( P 0 ) ≥ P 0 / ( 3 ln P 0 ) \pi(2P_0)-\pi(P_0)\ge P_0/(3\ln P_0) π ( 2 P 0 ) − π ( P 0 ) ≥ P 0 / ( 3 ln P 0 ) P 0 ≥ 100 P_0\ge100 P 0 ≥ 100 0.483 0.483 0.483 P 0 = 100 P_0=100 P 0 = 100 [ 100 , 2 ⋅ 10 4 ] [100,2\cdot10^4] [ 100 , 2 ⋅ 1 0 4 ] 10 5 10^5 1 0 5 10 6 10^6 1 0 6 10 7 10^7 1 0 7
Proposition K′. Let D e q = E # { ( τ , τ ′ ) ∈ P 2 , τ ≠ τ ′ , both hitting a , S y = S y ′ } D_{\mathrm{eq}}=\mathbb{E}\#\{(\tau,\tau')\in\mathcal{P}^2,\ \tau\ne\tau',\text{ both hitting }a,\ S_y=S_{y'}\} D eq = E # {( τ , τ ′ ) ∈ P 2 , τ = τ ′ , both hitting a , S y = S y ′ }
E [ P ⋅ Y P ] ≤ 2 P 0 ( E N + D e q ) + 6 ln ( m M ) ( E N ) 2 , \mathbb{E}[P\cdot Y_P]\le2P_0(\mathbb{E}N+D_{\mathrm{eq}})+6\ln(mM)(\mathbb{E}N)^2, E [ P ⋅ Y P ] ≤ 2 P 0 ( E N + D eq ) + 6 ln ( m M ) ( E N ) 2 , and hence
Pr ( round succeeds ) ≥ [ 2 P 0 E N ( 1 + D e q E N ) + 6 ln ( m M ) E N 2 ( E N ) 2 ] − 1 . \Pr(\text{round succeeds}) \ge\left[\frac{2P_0}{EN}\left(1+\frac{D_{\mathrm{eq}}}{EN}\right)+6\ln(mM)\frac{EN^2}{(EN)^2}\right]^{-1}. Pr ( round succeeds ) ≥ [ E N 2 P 0 ( 1 + E N D eq ) + 6 ln ( m M ) ( E N ) 2 E N 2 ] − 1 . Proof.
Lemma V (equal half-sums). For P 1 \mathcal{P}_1 P 1 ψ = 2 m / 2 / M \psi=2^{m/2}/M ψ = 2 m /2 / M M ≥ 2 M\ge2 M ≥ 2 m ≥ m 0 ( g ) m\ge m_0(g) m ≥ m 0 ( g )
D e q λ ≤ 0.72 e g 2 / 2 m ( 3 ψ + ψ 2 ) ( 1 + O ( m − 1 / 5 ) ) , λ = E [ Z ( u ) ∣ S E = s ] . \frac{D_{\mathrm{eq}}}{\lambda}\le0.72e^{g^2/2}m(3\psi+\psi^2)\left(1+O(m^{-1/5})\right),\qquad\lambda=E[Z(u)\mid S_E=s]. λ D eq ≤ 0.72 e g 2 /2 m ( 3 ψ + ψ 2 ) ( 1 + O ( m − 1/5 ) ) , λ = E [ Z ( u ) ∣ S E = s ] . Proof. Three facts are used.
(T2) Let X 1 , … , X K X_1,\ldots,X_K X 1 , … , X K Λ \Lambda Λ d × K d\times K d × K d d d Pr ( Λ X = b ) ≤ ∏ t max v Pr ( X j t = v ) \Pr(\Lambda X=b)\le\prod_t\max_v\Pr(X_{j_t}=v) Pr ( Λ X = b ) ≤ ∏ t max v Pr ( X j t = v )
A pair ( x , x ′ ) (x,x') ( x , x ′ ) E E E x x x x ′ x' x ′ { S x = a , S x ′ = a , S x A = S x A ′ , S E = s } \{S_x=a,S_{x'}=a,S_{x_A}=S_{x'_A},S_E=s\} { S x = a , S x ′ = a , S x A = S x A ′ , S E = s } ≥ 1 \ge1 ≥ 1
Otherwise x ′ = E ∖ x x' = E \setminus x x ′ = E ∖ x u = 0 u = 0 u = 0 S E = s S_E = s S E = s S x = a S_x = a S x = a ( 10 A , 10 B , 01 A ) (10_A,10_B,01_A) ( 1 0 A , 1 0 B , 0 1 A ) ≤ M − 3 \le M^{-3} ≤ M − 3 2 m 2^m 2 m
Summing. D e q ≤ [ ( 2 2 m − h + 2 m + h + 2 m ) M − 3 + 4 m M − 4 ] / Pr ( S E = s ) D_{\mathrm{eq}} \le[(2^{2m-h} + 2^{m+h} + 2^m)M^{-3} + 4^mM^{-4}]/\Pr(S_E = s) D eq ≤ [( 2 2 m − h + 2 m + h + 2 m ) M − 3 + 4 m M − 4 ] / Pr ( S E = s ) λ = λ m ( 1 + O ( m − 1 / 5 ) ) \lambda= \lambda_m(1 + O(m^{-1/5})) λ = λ m ( 1 + O ( m − 1/5 )) λ m = 2 m + 1 / ( M 2 π m c M ) \lambda_m = 2^{m+1}/(M\sqrt{2\pi m c_M}) λ m = 2 m + 1 / ( M 2 π m c M ) c M ≤ 5 / 8 c_M \le5/8 c M ≤ 5/8 σ < M / 12 \sigma< M/\sqrt{12} σ < M / 12
D e q λ ≤ ( 1 + O ( m − 1 / 5 ) ) 2 π c M 2 12 c 1 m [ 2 m − h + 2 h + 1 M + 2 m M 2 ] . \frac{D_{\mathrm{eq}}}{\lambda} \le(1 + O(m^{-1/5})) \frac{\sqrt{2\pi c_M}}{2\sqrt{12}c_1} m \left[\frac{2^{m-h} + 2^h + 1}{M} + \frac{2^m}{M^2}\right]. λ D eq ≤ ( 1 + O ( m − 1/5 )) 2 12 c 1 2 π c M m [ M 2 m − h + 2 h + 1 + M 2 2 m ] . Finally 2 m − h + 2 h + 1 ≤ 3 ⋅ 2 m / 2 2^{m-h} + 2^h + 1 \le3 \cdot2^{m/2} 2 m − h + 2 h + 1 ≤ 3 ⋅ 2 m /2 2 π ⋅ 5 / 8 ⋅ 2 π / ( 2 12 ) ≤ 0.72 \sqrt{2\pi}\cdot5/8\cdot\sqrt{2\pi}/(2\sqrt{12}) \le0.72 2 π ⋅ 5/8 ⋅ 2 π / ( 2 12 ) ≤ 0.72
Theorem R′′ (filtered endgame). Take the setting of Theorem R′ with R R R r r r ∣ T − S / 2 ∣ ≤ S G / 2 − R m M \lvert T - S/2\rvert\le S_G/2 - RmM ∣ T − S /2 ∣ ≤ S G /2 − R m M g ≥ 0 g \ge0 g ≥ 0 2 e − g 2 / 6 < 1 2e^{-g^2/6} < 1 2 e − g 2 /6 < 1 g = 3 g = 3 g = 3 j j j § 7.6 on E j E_j E j P j P_j P j [ P 0 , 2 P 0 ) [P_0,2P_0) [ P 0 , 2 P 0 ) r j r_j r j Z / P j \mathbb{Z}/P_j Z / P j P 0 ≥ 100 P_0 \ge100 P 0 ≥ 100
Then, given the weights of G G G 1 − p 1-p 1 − p
p = [ 2 P 0 λ m ( 1 + 0.72 e g 2 / 2 m ( 3 ψ + ψ 2 ) ) + 6 ln ( m M ) ( 1 + 2 λ m + O ( m − 1 / 5 ) ) ] − 1 ( 1 − O ( m − 1 / 5 ) ) . p = \left[\frac{2P_0}{\lambda_m}\left(1 + 0.72e^{g^2/2}m(3\psi+ \psi^2)\right) + 6\ln(mM)\left(1 + \frac{2}{\lambda_m} + O(m^{-1/5})\right)\right]^{-1}(1 - O(m^{-1/5})). p = [ λ m 2 P 0 ( 1 + 0.72 e g 2 /2 m ( 3 ψ + ψ 2 ) ) + 6 ln ( m M ) ( 1 + λ m 2 + O ( m − 1/5 ) ) ] − 1 ( 1 − O ( m − 1/5 )) . The expected cost of a round is O ( m ⋅ 2 m / 4 + m ⋅ 2 m / 2 / P 0 ) O(m \cdot2^{m/4} + m \cdot2^{m/2}/P_0) O ( m ⋅ 2 m /4 + m ⋅ 2 m /2 / P 0 )
A round whose block sum leaves the window counts as failed; this happens with probability at most 2 e − g 2 / 6 2e^{-g^2/6} 2 e − g 2 /6
Now choose P 0 = ⌊ λ m / 2 ⌋ P_0 = \lfloor\lambda_m/2\rfloor P 0 = ⌊ λ m /2 ⌋ m m m 2 m ≥ ( σ m ) 4 / 3 2^m \ge(\sigma\sqrt{m})^{4/3} 2 m ≥ ( σ m ) 4/3 2 m ≤ 2 ( σ m ) 4 / 3 2^m \le2(\sigma\sqrt{m})^{4/3} 2 m ≤ 2 ( σ m ) 4/3 m = ( 4 / 3 ) b + ( 2 / 3 ) log 2 b + O ( 1 ) m = (4/3)b + (2/3)\log_2 b + O(1) m = ( 4/3 ) b + ( 2/3 ) log 2 b + O ( 1 )
with R = ⌈ ln ( 1 / δ ) / ( ( 1 − 2 e − g 2 / 6 ) ⋅ p ) ⌉ R = \lceil\ln(1/\delta)/((1 - 2e^{-g^2/6})\cdot p)\rceil R = ⌈ ln ( 1/ δ ) / (( 1 − 2 e − g 2 /6 ) ⋅ p )⌉ ∣ T − S / 2 ∣ ≤ S G / 2 − R m M \lvert T - S/2\rvert\le S_G/2 - RmM ∣ T − S /2 ∣ ≤ S G /2 − R m M δ \delta δ n ≥ ∣ G ∣ + R m n \ge\lvert G\rvert+ Rm n ≥ ∣ G ∣ + R m
Proof.
As in Theorem R′, the weights of G G G
In a round, Proposition K′ applies. By Lemma B, E N = λ = λ m ( 1 + O ( m − 1 / 5 ) ) \mathbb{E}N = \lambda= \lambda_m(1 + O(m^{-1/5})) E N = λ = λ m ( 1 + O ( m − 1/5 )) E N 2 ≤ λ + λ ⋅ 1 [ u = 0 ] + λ 2 ( 1 + O ( m − 1 / 5 ) ) \mathbb{E}N^2 \le\lambda+ \lambda\cdot1[u = 0] + \lambda^2(1 + O(m^{-1/5})) E N 2 ≤ λ + λ ⋅ 1 [ u = 0 ] + λ 2 ( 1 + O ( m − 1/5 )) D e q D_{\mathrm{eq}} D eq p p p
The filtered lists have expected size 2 ∣ A ∣ ⋅ E [ 1 / P ] 2^{|A|}\cdot\mathbb{E}[1/P] 2 ∣ A ∣ ⋅ E [ 1/ P ] 2 ∣ B ∣ ⋅ E [ 1 / P ] 2^{|B|}\cdot\mathbb{E}[1/P] 2 ∣ B ∣ ⋅ E [ 1/ P ] O ( 2 m / 2 / P 0 ) O(2^{m/2}/P_0) O ( 2 m /2 / P 0 ) r r r P P P O ( 2 m / 4 ) O(2^{m/4}) O ( 2 m /4 )
With the stated m m m 2 m / 4 ≍ ( σ m ) 1 / 3 2^{m/4} \asymp(\sigma\sqrt{m})^{1/3} 2 m /4 ≍ ( σ m ) 1/3 2 m / 2 / P 0 ≍ M m / 2 m / 2 ≍ ( σ m ) 1 / 3 2^{m/2}/P_0 \asymp M\sqrt{m}/2^{m/2} \asymp(\sigma\sqrt{m})^{1/3} 2 m /2 / P 0 ≍ M m / 2 m /2 ≍ ( σ m ) 1/3 λ m ≍ M 1 / 3 m 1 / 6 ≥ 200 \lambda_m \asymp M^{1/3}m^{1/6} \ge200 λ m ≍ M 1/3 m 1/6 ≥ 200 M M M m ψ ≍ m 4 / 3 M − 1 / 3 → 0 m\psi\asymp m^{4/3}M^{-1/3} \to0 m ψ ≍ m 4/3 M − 1/3 → 0
The block sums of disjoint blocks are independent, and Lemma A needs no window. By Hoeffding each block sum lies in the window with probability at least 1 − 2 e − g 2 / 6 1-2e^{-g^2/6} 1 − 2 e − g 2 /6 p p p ( 1 − ( 1 − 2 e − g 2 / 6 ) p ) R ≤ δ (1-(1-2e^{-g^2/6})p)^R \le\delta ( 1 − ( 1 − 2 e − g 2 /6 ) p ) R ≤ δ
Since g g g g g g p p p 6 ln ( m M ) ⋅ O ( m − 1 / 5 ) = o ( ln m M ) 6\ln(mM)\cdot O(m^{-1/5})=o(\ln mM) 6 ln ( m M ) ⋅ O ( m − 1/5 ) = o ( ln m M ) e g 2 / 2 m ψ → 0 e^{g^2/2m\psi}\to0 e g 2 /2 m ψ → 0 p ≥ ( 1 − o ( 1 ) ) / ( 6 ln m M ) p\ge(1-o(1))/(6\ln mM) p ≥ ( 1 − o ( 1 )) / ( 6 ln m M )
What is new, and what is not.
Theorem R″ adds a short proof for the filtered endgame as implemented in the theorem mode of § 7.6 , in the model of this paper; the practical solver, with its suffix block and repeated rounds, is outside it. Its constant 6 ln ( m M ) 6\ln(mM) 6 ln ( m M ) Table 9 the first round almost always succeeds. With P P P λ / 2 \lambda/2 λ /2 λ \lambda λ 1 / P 1/P 1/ P 2 / 3 2/3 2/3 P = P 0 P=P_0 P = P 0
Representations. Lemma J and Proposition K′ hold for any scheme, in particular for the representation schemes of [HGJ; BCJ]. In our endgame the density is a free parameter, and part of the multiplicity that representations create at density one is already supplied by the λ \lambda λ 2 b / 3 2^{b/3} 2 b /3 2 0.291 b 2^{0.291b} 2 0.291 b
Constraint propagation: structural collapse and convergence
The arcs read a set of weights through its quotients Z / q Z/q Z / q § 7 is built from a few operations: a forward pass that computes reachable sets, residue counts on quotients, injections of decisions, a backward pass, and a final search. This section develops the general theory of these operations on an arbitrary network of constraints; §8 reads the algorithm of § 7 inside it.
The idea in one line. A network of constraints splits into two parts. The structural part (equalities with parity, functional definitions on a DAG) is solved in one pass, without search. The computational part (assertions, cycles, non-bijective constraints read backwards) needs iteration to a fixpoint, and that is where the whole cost lives.
The model
Variables V V V W : V → P ( V a l ) W:V\to\mathcal{P}(\mathrm{Val}) W : V → P ( Val ) W ′ ⊑ W ⟺ ∀ v W ′ ( v ) ⊆ W ( v ) W'\sqsubseteq W\Longleftrightarrow\forall vW'(v)\subseteq W(v) W ′ ⊑ W ⟺ ∀ v W ′ ( v ) ⊆ W ( v ) s = ( facts , domains , clusters ) s=(\text{facts},\text{domains},\text{clusters}) s = ( facts , domains , clusters ) s ⊑ s ′ ⟺ F ( s ) ⊆ F ( s ′ ) ∧ ∀ v D s ( v ) ⊆ D s ′ ( v ) s\sqsubseteq s'\Longleftrightarrow F(s)\subseteq F(s')\land\forall vD_s(v)\subseteq D_{s'}(v) s ⊑ s ′ ⟺ F ( s ) ⊆ F ( s ′ ) ∧ ∀ v D s ( v ) ⊆ D s ′ ( v )
Projection π v ( c , W ) = { a ∈ W ( v ) ∣ ∃ α ( α ( v ) = a ∧ c ( α ) ∧ ∀ u ∈ vars ( c ) α ( u ) ∈ W ( u ) } \pi_v(c,W)=\{a\in W(v)\mid\exists\alpha(\alpha(v)=a\land c(\alpha)\land\forall u\in\operatorname{vars}(c)\alpha(u)\in W(u)\} π v ( c , W ) = { a ∈ W ( v ) ∣ ∃ α ( α ( v ) = a ∧ c ( α ) ∧ ∀ u ∈ vars ( c ) α ( u ) ∈ W ( u )} T c ( W ) ( v ) = π v ( c , W ) T_c(W)(v)=\pi_v(c,W) T c ( W ) ( v ) = π v ( c , W ) υ ( c , W ) = max v ∣ W ( v ) ∖ π v ( c , W ) ∣ / ∣ W ( v ) ∣ \upsilon(c,W)=\max_v |W(v)\setminus\pi_v(c,W)|/|W(v)| υ ( c , W ) = max v ∣ W ( v ) ∖ π v ( c , W ) ∣/∣ W ( v ) ∣ υ \upsilon υ o ( W ) o(W) o ( W )
The structural part
Definition 7.1. T s t r u c t = Eval ∘ Par T_{\mathrm{struct}}=\operatorname{Eval}\circ\operatorname{Par} T struct = Eval ∘ Par
Par is union-find with parity on C p a r C_{\mathrm{par}} C par x = r ⊕ b x x=r\oplus b_x x = r ⊕ b x r r r S = { W ∣ ∀ x ∼ y , W ( x ) = W ( y ) ⊕ ( b x ⊕ b y ) } \mathcal{S}=\{W\mid\forall x\sim y,W(x)=W(y)\oplus(b_x\oplus b_y)\} S = { W ∣ ∀ x ∼ y , W ( x ) = W ( y ) ⊕ ( b x ⊕ b y )}
Eval is one pass over Def in a topological order of the parity classes (the graph of the definitions with the classes of Par contracted, assumed acyclic): W ( x ) ← W ( x ) ∩ f x ( ∏ u ∈ pred ( x ) W ( u ) ) W(x)\leftarrow W(x)\cap f_x(\prod_{u\in\operatorname{pred}(x)}W(u)) W ( x ) ← W ( x ) ∩ f x ( ∏ u ∈ pred ( x ) W ( u ))
Theorem 7.2 (structural collapse).
(a) T s t r u c t ∘ T s t r u c t = T s t r u c t T_{\mathrm{struct}}\circ T_{\mathrm{struct}}=T_{\mathrm{struct}} T struct ∘ T struct = T struct T s t r u c t ( W ) ∈ S T_{\mathrm{struct}}(W)\in\mathcal{S} T struct ( W ) ∈ S
(b) For W ∈ S W\in\mathcal{S} W ∈ S T s t r u c t ( W ) = W T_{\mathrm{struct}}(W)=W T struct ( W ) = W
(c) Injection: if W ∈ S W\in\mathcal{S} W ∈ S O ( 1 ) O(1) O ( 1 ) O ( ∑ x cost ( f x ) ) O(\sum_x\operatorname{cost}(f_x)) O ( ∑ x cost ( f x )) α \alpha α α ( r ) \alpha(r) α ( r ) { α ( x ) } \{\alpha(x)\} { α ( x )} x : = u x:=u x := u y : = ¬ u y:=\neg u y := ¬ u x = y x=y x = y O ( 1 ) O(1) O ( 1 )
(d) For W ∈ S W\in\mathcal{S} W ∈ S υ ( c , W ) = 0 \upsilon(c,W)=0 υ ( c , W ) = 0 c ∈ C p a r c\in C_{\mathrm{par}} c ∈ C par
Proof. (a) On a parity class the intersection, with the bit applied, is the same for every member, and intersecting again changes nothing. In a topological order of the classes every class receives the images of parents whose classes are already final, so a second pass changes nothing. (b) Follows from (a). (c) An injection into S \mathcal{S} S f x ( singletons ) f_x(\text{singletons}) f x ( singletons ) α \alpha α α ( r ) \alpha(r) α ( r ) { α ( x ) } \{\alpha(x)\} { α ( x )} S \mathcal{S} S x x x
Scope. The collapse holds on the shared form, not for every W ′ ≥ T s t r u c t ( W 0 ) W' \ge T_{\mathrm{struct}}(W_0) W ′ ≥ T struct ( W 0 )
Remark 7.2′ (the order of Eval). With x : = u x := u x := u z : = w z := w z := w u = z u = z u = z x , z x, z x , z z z z u u u x x x x x x T s t r u c t T_{\mathrm{struct}} T struct Tstruct_not_idem_counterexample).
Theorem 7.3 (information limit). Let S S S v v v ∣ S ∣ ≤ 2 |S| \le2 ∣ S ∣ ≤ 2 V V V ∣ S ∣ ≤ 2 ∣ I ∣ |S| \le2^{|I|} ∣ S ∣ ≤ 2 ∣ I ∣
Proof. If fixing v v v s ↦ s v s \mapsto s_v s ↦ s v S S S S ′ S' S ′ S S S ∣ S ∣ ≤ ∣ S ′ ∣ ≤ 2 |S| \le|S'| \le2 ∣ S ∣ ≤ ∣ S ′ ∣ ≤ 2 Theorem 7.2 (c)). □
Reading. “Solve one, solve all” holds exactly on the components with a single free input. An auxiliary variable X X X o ≤ 1 o \le1 o ≤ 1 Lemma 2.1 ) is the case in which the certificate is built backwards in O ( n ) O(n) O ( n ) § 3 ).
The computational part
Theorem 7.4. Let V V V T c T_c T c c ∈ C c o m p ∪ D e f c \in\mathcal{C}_{\mathrm{comp}} \cup\mathrm{Def} c ∈ C comp ∪ Def
(i) T c o m p T_{\mathrm{comp}} T comp T c o m p ( W ) ⊑ W T_{\mathrm{comp}}(W) \sqsubseteq W T comp ( W ) ⊑ W W ⊑ W ′ → T c o m p ( W ) ⊑ T c o m p ( W ′ ) W \sqsubseteq W' \to T_{\mathrm{comp}}(W) \sqsubseteq T_{\mathrm{comp}}(W') W ⊑ W ′ → T comp ( W ) ⊑ T comp ( W ′ )
(ii) T c o m p ( W ) = W ⟺ ∀ c ν ( c , W ) = 0 T_{\mathrm{comp}}(W) = W \Longleftrightarrow\forall c\,\nu(c,W) = 0 T comp ( W ) = W ⟺ ∀ c ν ( c , W ) = 0
(iii) Let μ ( W ) = ∑ v ( ∣ W ( v ) ∣ − 1 ) \mu(W) = \sum_v (|W(v)| - 1) μ ( W ) = ∑ v ( ∣ W ( v ) ∣ − 1 ) T f a i l T_{\mathrm{fail}} T fail T c o m p T_{\mathrm{comp}} T comp W W W T f a i l m ( W ) = T f a i l μ ( W ) + 1 ( W ) T_{\mathrm{fail}}^m(W) = T_{\mathrm{fail}}^{\mu(W)+1}(W) T fail m ( W ) = T fail μ ( W ) + 1 ( W ) m ≥ μ ( W ) + 1 m \ge\mu(W) + 1 m ≥ μ ( W ) + 1 μ ( W ) \mu(W) μ ( W ) W = { 0 } W = \{0\} W = { 0 } x = 1 x = 1 x = 1 μ ( W ) = 0 \mu(W) = 0 μ ( W ) = 0 T c o m p ( W ) ≠ W T_{\mathrm{comp}}(W) \ne W T comp ( W ) = W Tcomp_bound_counterexample).
(iv) The limit W ∗ W^* W ∗ W ′ ⊑ W W' \sqsubseteq W W ′ ⊑ W ν ( c , W ′ ) = 0 \nu(c,W') = 0 ν ( c , W ′ ) = 0 c c c
Proof. (i) Each filter shrinks domains, and projection is monotone. (ii) T c ( W ) = W ⟺ π v ( c , W ) = W ( v ) T_c(W) = W \Longleftrightarrow\pi_v(c,W) = W(v) T c ( W ) = W ⟺ π v ( c , W ) = W ( v ) μ ∈ N \mu\in\mathbb{N} μ ∈ N W ′ ⊑ W W' \sqsubseteq W W ′ ⊑ W T c T_c T c
Remark 7.5 (a natural bound that fails). A natural bound on the number of iterations is ∑ c ν ( c , W 0 ) ⋅ d \sum_c \nu(c,W_0) \cdot d ∑ c ν ( c , W 0 ) ⋅ d x 1 ≠ ⋯ ≠ x k x_1 \ne\cdots\ne x_k x 1 = ⋯ = x k D ( x 1 ) = { 1 } D(x_1)=\{1\} D ( x 1 ) = { 1 } D ( x i ) = { i − 1 , i } D(x_i)=\{i-1,i\} D ( x i ) = { i − 1 , i } ∑ c υ ( c , W 0 ) ⋅ d = 1 \sum_c \upsilon(c,W_0)\cdot d=1 ∑ c υ ( c , W 0 ) ⋅ d = 1 k − 1 k-1 k − 1 k = 3 , 5 , 10 , 20 k=3,5,10,20 k = 3 , 5 , 10 , 20 μ ( W ) \mu(W) μ ( W ) Theorem 7.4 (iii) has the right order.
Factorization and algorithm
Theorem 7.6 (factorization). T = T c o m p ∘ T s t r u c t T=T_{\mathrm{comp}}\circ T_{\mathrm{struct}} T = T comp ∘ T struct
Par costs O ( ∣ C p a r ∣ ⋅ α ( n ) ) O(|C_{\mathrm{par}}|\cdot\alpha(n)) O ( ∣ C par ∣ ⋅ α ( n )) O ( ∑ x cost ( f x ) ) O(\sum_x \operatorname{cost}(f_x)) O ( ∑ x cost ( f x ))
Every injection costs O ( 1 ) O(1) O ( 1 )
T c o m p T_{\mathrm{comp}} T comp μ ( W ) + 1 \mu(W)+1 μ ( W ) + 1 W ∗ W^* W ∗
Algorithm.
Union-find with parity on C p a r C_{\mathrm{par}} C par
Topological order of the parity classes of the definitions (complete DAG with the classes contracted, active arcs).
Injections: update the class, then Eval forwards.
AC-3-style worklist [Mackworth] on C c o m p C_{\mathrm{comp}} C comp
Outcome: an empty domain means unsatisfiable; all singletons means a solution; otherwise the network is locally consistent and search over the free inputs is needed (at most 2 ∣ I ∣ 2^{|I|} 2 ∣ I ∣ Theorem 7.3 ).
Definitional variables
The aim is to determine as much as possible in one pass: once one variable (or a few) is fixed, how many others follow without search. The tool is the definitional variable X : = f ( nodes ) X:=f(\text{nodes}) X := f ( nodes )
Theorem 7.7. Let S S S R A B = π A , B ( S ) R_{AB}=\pi_{A,B}(S) R A B = π A , B ( S ) X X X A A A B B B T s t r u c t T_{\mathrm{struct}} T struct R A B R_{AB} R A B B = g ( A ) B=g(A) B = g ( A ) S S S A A A B B B g g g
Proof. (⇐ \Leftarrow ⇐ X : = g ( A ) X:=g(A) X := g ( A ) B = X B=X B = X A A A X X X B B B ⇒ \Rightarrow ⇒ A A A B B B B B B A A A Theorem 7.3 ). □
X X X R A B R_{AB} R A B A ∨ B = 1 A\lor B=1 A ∨ B = 1 X X X
Theorem 7.7 is the network form of implicit definability: Beth’s theorem turns implicit into explicit definability, Padoa’s method tests it, and Craig interpolation extracts X X X
Algorithm Ext. Repeat until nothing changes:
Linear X X X Each XOR constraint, of any arity, and each known value is a row. The reduced form gives, for every pivot, X : = ⨁ ( free variables ) ⊕ b X := \bigoplus(\text{free variables}) \oplus b X := ⨁ ( free variables ) ⊕ b
Local X X X With one input known, a definition becomes linear: and ( a , 1 ) ⇒ y = a \operatorname{and}(a,1) \Rightarrow y=a and ( a , 1 ) ⇒ y = a and ( a , 0 ) ⇒ y = 0 \operatorname{and}(a,0) \Rightarrow y=0 and ( a , 0 ) ⇒ y = 0 ite ( 1 , a , b ) ⇒ y = a \operatorname{ite}(1,a,b) \Rightarrow y=a ite ( 1 , a , b ) ⇒ y = a ite ( c , a , a ) ⇒ y = a \operatorname{ite}(c,a,a) \Rightarrow y=a ite ( c , a , a ) ⇒ y = a not ( a ) ⇒ y = ¬ a \operatorname{not}(a) \Rightarrow y=\neg a not ( a ) ⇒ y = ¬ a
Arc consistency on the rest (definitions not yet linear, also read backwards). New values return to step 1.
Proposition 7.8 (soundness). Every value fixed by Ext holds in every solution. Proof. The rows of step 2 are implied by the definitions with the known input; Gauss and arc consistency are sound; induction on the rounds. □
Proposition 7.9 (completeness in the affine case). If the network has only XOR constraints, Ext determines exactly the forced variables, and for every pair ( A , B ) (A,B) ( A , B ) B B B A A A R A B R_{AB} R A B Proof. The solutions form an affine subspace. x x x ⟺ e x \Longleftrightarrow e_x ⟺ e x ⟺ \Longleftrightarrow ⟺ e x e_x e x x x x
This is the affine class of Schaefer’s dichotomy [Schaefer], and Gaussian elimination for XOR constraints is standard in SAT solving [SNC]. The point here is the reading: Gauss is the systematic construction of the X X X Theorem 7.7 , and it moves every XOR constraint from the computational to the structural part.
Probing: non-local X X X For each free variable A A A A = 0 A=0 A = 0 A = 1 A=1 A = 1
If one branch fails, A A A
If B B B B B B
If B B B A A A X X X B = A ⊕ c B=A\oplus c B = A ⊕ c
The new rows return to Ext, and the procedure is repeated until nothing changes. One round costs 2 n 2n 2 n X X X
Proposition 7.10 (soundness of probing). Every X X X Proof. Every solution has A = 0 A=0 A = 0 A = 1 A=1 A = 1 A A A B B B
Proposition 7.10′ (probing to depth ℓ \ell ℓ Let Probe 0 = Ext \operatorname{Probe}_0 = \operatorname{Ext} Probe 0 = Ext Probe ℓ \operatorname{Probe}_{\ell} Probe ℓ Probe ℓ − 1 \operatorname{Probe}_{\ell-1} Probe ℓ − 1 A = 0 A = 0 A = 0 A = 1 A = 1 A = 1 A A A
Then:
(a) every X X X Probe ℓ \operatorname{Probe}_{\ell} Probe ℓ
(b) if each inner Probe ℓ − 1 \operatorname{Probe}_{\ell-1} Probe ℓ − 1 Probe ℓ \operatorname{Probe}_{\ell} Probe ℓ ( 2 n ) ℓ (2n)^{\ell} ( 2 n ) ℓ Ext \operatorname{Ext} Ext
(c) if some set B B B ℓ \ell ℓ Ext \operatorname{Ext} Ext B B B Ext \operatorname{Ext} Ext Probe ℓ \operatorname{Probe}_{\ell} Probe ℓ
Proof. (a) Induction on ℓ \ell ℓ ℓ = 0 \ell= 0 ℓ = 0 Proposition 7.8 , and the step is the argument of Proposition 7.10 with Probe ℓ − 1 \operatorname{Probe}_{\ell-1} Probe ℓ − 1 Ext \operatorname{Ext} Ext 2 n 2n 2 n Probe ℓ − 1 \operatorname{Probe}_{\ell-1} Probe ℓ − 1 ℓ \ell ℓ ℓ = 0 \ell= 0 ℓ = 0 B = ∅ B = \varnothing B = ∅ Ext \operatorname{Ext} Ext ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 A ∈ B A \in B A ∈ B B ∖ { A } B \setminus\{A\} B ∖ { A } ℓ − 1 \ell- 1 ℓ − 1 Probe ℓ \operatorname{Probe}_{\ell} Probe ℓ X X X c c c Probe ℓ − 1 \operatorname{Probe}_{\ell-1} Probe ℓ − 1 X = c X = c X = c Probe ℓ \operatorname{Probe}_{\ell} Probe ℓ X = c X = c X = c
One-level probing cannot be complete in general (it would decide SAT in polynomial time); probing to depth ℓ \ell ℓ ( 2 n ) ℓ (2n)^{\ell} ( 2 n ) ℓ ℓ \ell ℓ Proposition 7.10 ′), and beyond them search remains. The three levels (Gauss, local linearization, probing) all produce correct definitional X X X Theorem 7.7 ). The limit of what can be determined in one pass is exactly the functionality of R A B R_{AB} R A B
Convergence of the full operator
The propagation operator acts on states s = ( F , D , Cl ) s = (F,D,\operatorname{Cl}) s = ( F , D , Cl ) K = Merge ∘ Unary ∘ Bridge ∘ LocalDed K = \operatorname{Merge} \circ\operatorname{Unary} \circ\operatorname{Bridge} \circ\operatorname{LocalDed} K = Merge ∘ Unary ∘ Bridge ∘ LocalDed
LocalDed applies inference rules inside each cluster;
Bridge keeps a value of a variable only if some consistent assignment satisfies, together, all the constraints touching that variable (a filter stronger than arc consistency, close to a global solve on a complete graph);
Unary applies the unary constraints;
Merge merges clusters that share variables.
Its measure is ν ( s ) = ∣ Cl ( s ) ∣ + ∑ v ∣ D s ( v ) ∣ \nu(s) = |\operatorname{Cl}(s)| + \sum_v |D_s(v)| ν ( s ) = ∣ Cl ( s ) ∣ + ∑ v ∣ D s ( v ) ∣
Lemma 7.11.
L1: F ( K s ) = F ( s ) F(Ks) = F(s) F ( K s ) = F ( s ) D K s ( v ) ⊆ D s ( v ) D_{Ks}(v) \subseteq D_s(v) D K s ( v ) ⊆ D s ( v )
L3: Cl ( K s ) = Cl ( s ) \operatorname{Cl}(Ks) = \operatorname{Cl}(s) Cl ( K s ) = Cl ( s ) ∣ Cl ( K s ) ∣ < ∣ Cl ( s ) ∣ |\operatorname{Cl}(Ks)| < |\operatorname{Cl}(s)| ∣ Cl ( K s ) ∣ < ∣ Cl ( s ) ∣
laps returns the same clusters; with an overlap the count drops.)
Theorem 7.12 (termination). ∀ s 0 ∃ n ≤ ν ( s 0 ) ∀ m ≥ n K m ( s 0 ) = K n ( s 0 ) \forall s_0 \exists n \le\nu(s_0) \forall m \ge n K^m(s_0) = K^n(s_0) ∀ s 0 ∃ n ≤ ν ( s 0 ) ∀ m ≥ n K m ( s 0 ) = K n ( s 0 ) Proof. By L4, as long as there is no fixpoint ν \nu ν N \mathbb{N} N
Theorem 7.13 (closure). With s ∗ = K n ( s 0 ) s^* = K^n(s_0) s ∗ = K n ( s 0 ) Fix ( s ∗ ) ∧ s 0 ⪯ s ∗ ∧ ∀ s ′ ( Fix ( s ′ ) ∧ s 0 ⪯ s ′ → s ∗ ⪯ s ′ ) \operatorname{Fix}(s^*) \land s_0 \preceq s^* \land\forall s'(\operatorname{Fix}(s') \land s_0 \preceq s' \to s^* \preceq s') Fix ( s ∗ ) ∧ s 0 ⪯ s ∗ ∧ ∀ s ′ ( Fix ( s ′ ) ∧ s 0 ⪯ s ′ → s ∗ ⪯ s ′ ) s 0 s_0 s 0 Proof. By induction on k k k K k ( s 0 ) ⪯ s ′ K^k(s_0) \preceq s' K k ( s 0 ) ⪯ s ′ Fix ( s ′ ) \operatorname{Fix}(s') Fix ( s ′ ) k = n k=n k = n s ∗ ⪯ s ′ s^* \preceq s' s ∗ ⪯ s ′
Remark 7.14 (a maximality statement that fails). The natural statement that a maximal fixpoint is reached is false. With one domain { 0 , 1 } \{0,1\} { 0 , 1 } { 0 } \{0\} { 0 } ( t , g , 0 ) (t,g,0) ( t , g , 0 ) s ∗ s^* s ∗
Tests
All tests compare with brute force on small networks; scripts are listed in Data and code availability.
Table 18.
Scale (XOR constraints with a planted solution, 1% of the variables injected): n = 1000 n = 1000 n = 1000 n = 2000 n = 2000 n = 2000
The Subset Sum algorithm as propagation
Model. Subset Sum is the network of §7.1 with inputs x i ∈ { 0 , 1 } x_i \in\{0,1\} x i ∈ { 0 , 1 } s 0 = 0 s_0 = 0 s 0 = 0 s i : = s i − 1 + w i ⋅ x i s_i := s_{i-1} + w_i \cdot x_i s i := s i − 1 + w i ⋅ x i s n = T s_n = T s n = T
Proposition 8.1 (forward pass = Eval). The forward pass R i = R i − 1 ∪ ( R i − 1 + w i ) R_i = R_{i-1} \cup(R_{i-1} + w_i) R i = R i − 1 ∪ ( R i − 1 + w i ) R i = W ( s i ) R_i = W(s_i) R i = W ( s i ) Proof. Forward image of s i − 1 + w i ⋅ x i s_{i-1} + w_i \cdot x_i s i − 1 + w i ⋅ x i x i ∈ { 0 , 1 } x_i \in\{0,1\} x i ∈ { 0 , 1 }
Proposition 8.2 (one backward round suffices). After Eval, one round of the arc consistency filters from i = n i = n i = n W ∗ W^* W ∗ T c o m p T_{\mathrm{comp}} T comp W ∗ ( s i ) W^*(s_i) W ∗ ( s i ) T T T w i w_i w i T − w i ∈ R i − 1 T - w_i \in R_{i-1} T − w i ∈ R i − 1
Proof. Downward induction on i i i W ∗ ( s i ) W^*(s_i) W ∗ ( s i ) i i i s i − 1 s_{i-1} s i − 1 v ∈ R i − 1 v \in R_{i-1} v ∈ R i − 1 v ∈ W ∗ ( s i ) v \in W^*(s_i) v ∈ W ∗ ( s i ) v + w i ∈ W ∗ ( s i ) v + w_i \in W^*(s_i) v + w i ∈ W ∗ ( s i ) i i i
This is a case of Freuder’s theorem on acyclic networks [Freuder]. The bound μ ( W ) \mu(W) μ ( W ) n n n K i K_i K i T = T c o m p ∘ T s t r u c t T = T_{\mathrm{comp}} \circ T_{\mathrm{struct}} T = T comp ∘ T struct O ( n ⋅ K max ) O(n \cdot K_{\max}) O ( n ⋅ K m a x ) o ≤ 1 o \le1 o ≤ 1 K = 1 K = 1 K = 1 O ( n ) O(n) O ( n )
Proposition 8.3 (the dummies are an extension that does not block but does not conserve). W ′ = W ∪ D W' = W \cup D W ′ = W ∪ D K = 1 K = 1 K = 1 R ( W ′ ) ⊇ R ( W ) R(W') \supseteq R(W) R ( W ′ ) ⊇ R ( W ) W ′ W' W ′ T ∉ R ( W ′ ) ⟹ T ∉ R ( W ) T \notin R(W') \Longrightarrow T \notin R(W) T ∈ / R ( W ′ ) ⟹ T ∈ / R ( W )
The exact table R i R_i R i n ⋅ K n \cdot K n ⋅ K O ( n + log max W ) O(n + \log\max W) O ( n + log max W ) ( n , log max W ) (n,\log\max W) ( n , log max W ) equivalent to a polynomial algorithm for Subset Sum. This is an equivalence, not progress.
Proposition 8.4 (the arcs are Eval on a quotient). N c ( q ) > 0 ⟺ N_c(q)>0 \Longleftrightarrow N c ( q ) > 0 ⟺ c c c q q q s i s_i s i Proof. N c ( q ) = # { subsets with sum ≡ c ( m o d q ) } N_c(q)=\#\{\text{subsets with sum}\equiv c\pmod q\} N c ( q ) = # { subsets with sum ≡ c ( mod q )} q q q □ \square □ Theorem 7.13 (a relaxation deletes no solution) it is always correct.
Three semirings (Remark 2.4). The chain is evaluated in three semirings, linked by homomorphisms:
Table 1 (PDF p. 55).
Propagation lives in B \mathbb{B} B N \mathbb{N} N C \mathbb{C} C B \mathbb{B} B ∏ ( 1 + z w i ) \prod(1+z^{w_i}) ∏ ( 1 + z w i )
Proposition 8.5 (the blur is invertible on counts, not on Booleans). With dummies 1 , 2 , … , 2 k − 1 1,2,\ldots,2^{k-1} 1 , 2 , … , 2 k − 1 M = 2 k M=2^k M = 2 k F W ∪ D = F W ⋅ ( 1 − z M ) / ( 1 − z ) F_{W\cup D}=F_W\cdot(1-z^M)/(1-z) F W ∪ D = F W ⋅ ( 1 − z M ) / ( 1 − z )
r W ( T ) = ∑ j ≥ 0 [ r W ∪ D ( T − j M ) − r W ∪ D ( T − j M − 1 ) ] . r_W(T)=\sum_{j\ge0}\left[r_{W\cup D}(T-jM)-r_{W\cup D}(T-jM-1)\right]. r W ( T ) = j ≥ 0 ∑ [ r W ∪ D ( T − j M ) − r W ∪ D ( T − j M − 1 ) ] . the blur identity of §5.4 solved by telescoping. In B [ Z ] \mathbb{B}[\mathbb{Z}] B [ Z ] { 0 , … , M − 1 } \{0,\ldots,M-1\} { 0 , … , M − 1 } R ( W ∪ D ) R(W\cup D) R ( W ∪ D ) R ( W ) R(W) R ( W )
The solver of §5 in the framework.
Table 2 (PDF p. 55).
Outside o ≤ 1 o \le1 o ≤ 1
Tests.
Table 15.
Discussion
Literature
Table 16.
In the dense regime Bringmann–Wellnitz is rigorous and faster. Our contribution there is the arc reading of their obstructions (almost divisors are the moduli with vanishing class counts), and the measured extension of the solver to weights of 10 10 − 10 12 10^{10}-10^{12} 1 0 10 − 1 0 12
For b = O ( ( log n ) 2 ) b=O((\log n)^2) b = O (( log n ) 2 ) b = n c b=n^c b = n c 2 O ( b / log n ) 2^{O(b/\log n)} 2 O ( b / l o g n ) 2 b / 2 + o ( b ) 2^{b/2+o(b)} 2 b /2 + o ( b )
Wan–Shi’s algorithm is the “no arcs, centre rule” variant of our ablation, stated heuristically and without experiments. Theorem R′ proves its guarantee. The arcs add the treatment of congruence obstructions (12 / 12 12/12 12/12 2 − 4 / 12 2-4/12 2 − 4/12
Theorem R″ is the four-list form [SS; Wag] of Theorem R′. It gives a proof for the theorem mode of the implemented endgame, through Lemma J, with the moments of Lemma B as the only probabilistic input.
On the number-theoretic side, Richert’s theorem (every integer > 6 >6 > 6 Π ( 1 + z p ) \Pi(1+z^p) Π ( 1 + z p )
On the propagation side (§ 9 –§ 10 ):
Union-find for equalities and forward evaluation of circuits are standard (congruence closure in SMT solvers, presolve in constraint programming). Labeled union-find [LUF] factors constraints by grouplabelled relations (affine, XOR, modular) and refines with an AC-3 schedule; it does not treat functional definitions or DAGs. What §7 adds is the single factorization T = T c o m p ∘ T s t r u c t T = T_{\mathrm{comp}} \circ T_{\mathrm{struct}} T = T comp ∘ T struct Theorem 7.3 ), the corrected bound (Theorem 7.4 (iii) and Remark 7.5), and the closure form of convergence (Theorem 7.13 ).
Counting subset sums by evaluating the generating polynomial at roots of unity is the method of [LN]. The residue classes mod q q q
Scope
The zeros. Propositions 3 and 30 are equivalences, and Theorem S is an Ω \Omega Ω
The Euler product. The Euler product can be read as a structural collapse of the coefficients onto the primes, with the sign of Re ζ ′ / ζ \operatorname{Re}\zeta'/\zeta Re ζ ′ / ζ
Conclusion
For the primes, completeness makes the decision problem trivial and the counting problem arithmetic. The generating function of the prime subset sums has arcs at the rationals a / q a/q a / q q = 2 k q=2^k q = 2 k q = 2 p q=2^p q = 2 p q q q ( R ) (R) ( R ) ( N ) (N) ( N ) q = 8 q=8 q = 8 ζ \zeta ζ
For arbitrary weights, the dummies of HURS restore completeness by blurring the counts, and the blur can be removed exactly, so the solver reads the arcs directly on the original weights. The arcs then become the decision rule of a solver whose cost is governed by its endgame: the square root of the omicron with a plain meet-in-the-middle, and about its cube root once the endgame is filtered by a residue class, which is itself an arc at a modulus of the order of the number of solutions. It returns only exact solutions, finds them on instances far beyond dynamic programming, and needs its arcs precisely where the structure theorem of dense Subset Sum puts the obstructions. On random weights of b b b 1 − δ 1-\delta 1 − δ 2 b / 2 + o ( b ) ⋅ log ( 1 / δ ) 2^{b/2+o(b)}\cdot\log(1/\delta) 2 b /2 + o ( b ) ⋅ log ( 1/ δ ) 2 b / 3 + o ( b ) 2^{b/3+o(b)} 2 b /3 + o ( b )
The algorithm is an instance of a general propagation theory. Every constraint network factors into a structural part solved in one pass and a computational part that converges, in any order, to the least informed fixpoint; definitional variables make explicit exactly the links that are functional on the solutions. Read in this theory, the forward pass of the solver is structural evaluation, the arcs are propagation on quotients, the dummies are an extension that does not block but does not conserve, and de-blurring works because that loss, irreversible on reachable sets, is reversible on counts. Dense instances in the sense of Bringmann–Wellnitz are the regime, beyond completeness, in which propagation alone decides.
Data and code availability
All computations are reproducible with the scripts listed here (Python, NumPy, SymPy, mpmath, matplotlib for the figures, python-flint/Arb for the certifications, and OR-Tools for the CP-SAT method of BENCH5; the versions used are listed in requirements.txt).
Arcs and races: python/arcs_theory/arcs_fig.py, arcform2.py, edgeworth_arcs.py, edgeworth_arcs3.py (Proposition C′, Table 2′), arc_races.py.
Lobes: python/arcs_theory/lobes2.py, lobe_proof_check.py, lobe_bound.py, lobe_tail.py; certification of Corollary Q′ in Arb: python/certification/certify_lobes_arb.py, with output python/certification/lobes_arb.txt.
Certification of a zero on the line for the cubic character mod 7 (Theorem S): python/certification/certify_zero_cubic7.py, a sign change of Z ( t ) Z(t) Z ( t ) python/certification/zero_cubic7_out128.txt.
Filtered endgame and floating-point counts (§5.4–5.6, Table 9 ′–Table 9 ″, Theorem R″): python/algorithm/arcs_float.py, endgame_fast.py, test_123.py, bench_big2.py, bench_small2.py.
The solver: python/algorithm/arcs_rho.py (default ARCS, parameters derived from the instance, §5.7) and python/algorithm/arcs_solver.py (ARCS with the fixed constants of BENCH5 v2), with the pseudocode in pseudocode/ARCS_pseudocode.md.
The benchmark of the default solver (§5.7): criterion with its sha256, runner, independent checker, results and logs, in benchmarks/arcs_rho/.
The benchmark BENCH5 (§5.7, Figure 9 ), in benchmarks/bench5_v2/. The reference run v2: criterion (with its sha256 and an English translation), bench5.py, checker5.py, export5.py, results, run and check logs, and an export of every instance and answer as plain text with verify_one.py, which verifies the YES answers (the NO certificates are verified by checker5.py).
Algorithm: python/algorithm/decider.py, constructor3.py, poisson.py, deblur.py, bench_deblur.py, bench_big.py, ablation.py, nearprime.py, reps.py; shared modules hurs_arcs.py and hurs_hybrid.py.
Section 6, in python/random_weights/: rig_check.py, b0_check.py, b0p_check.py, lemmaB_check.py, b_moment.py, b_cond.py, exact_B3.py, poisson_tail.py, moments4.py, b_prime_exact.py (Proposition B′), lv_rate.py and lv_rate_certified.py (Theorem LV).
Section 7, in python/propagation/: run_tests.py (Table 13 ), ext.py (algorithm Ext, Table 14 and the scale test), probe.py (probing).
Table 13. The generated networks have parity constraints between inputs only; the configuration of Remark 7.2 (a parity class with members defined at different depths) does not occur in them.
Table 14. Fraction of the variables forced by brute force (excluding the injected ones) that each method determines.
Figures, in python/figures/: Figure 5 comes from python/arcs_theory/arcs_fig.py; each of the others from its own script fig_*.py, with the shared style figstyle.py and, for Figure 6 , the measured times fig_endgame_rows.json.
Figure 5. The heights H a / q ( N ) ⋅ log N / N H_{a/q}(N)\cdot\log N/\sqrt{N} H a / q ( N ) ⋅ log N / N N N N 10 4 10^4 1 0 4 10 8 10^8 1 0 8 x x x ± 2 \pm2 ± 2 L ( 1 , χ ) L(1,\chi) L ( 1 , χ ) L ′ ( 1 , χ ) L'(1,\chi) L ′ ( 1 , χ ) Table 5 ). At q = 6 q=6 q = 6
Table 5. The races on the arcs against Rubinstein–Sarnak. With the finite-x x x 1 2 li ( x ) ⋅ log x / x \frac{1}{2}\operatorname{li}(\sqrt{x})\cdot\log x/\sqrt{x} 2 1 li ( x ) ⋅ log x / x ≈ 1.21 \approx1.21 ≈ 1.21 − 0.53 -0.53 − 0.53 − 0.29 -0.29 − 0.29 − 0.55 -0.55 − 0.55 − 0.29 -0.29 − 0.29 ± 2 \pm2 ± 2 Figure 5 ).
Figure 6. Time per block of the endgame on m m m λ ≈ 200 \lambda\approx200 λ ≈ 200 m ≈ 24 m \approx24 m ≈ 24 λ \lambda λ P P P 2 m / 2 2^{m/2} 2 m /2 P P P m = 44 m = 44 m = 44
A Lean 4 project, in lean/: the certified answer checker (checkYes, checkNo, checkRange, proved sound; the exhaustive certificate (− 3 , − 3 -3,-3 − 3 , − 3 verify_bench, which rechecks every answer of the two benchmarks; and the statements of the paper, including the analytic ones (Theorem E , Propositions C, C′, 3, 30, Theorem S, Lemmas L, W , Q, Corollary Q′, Lemmas B0′, B, B‴, V, Proposition B′, Theorems R, R′, R″, LV, and the propagation theory of §7–8): all 62 items of the paper are covered. Published theorems (Siegel–Walfisz, Vinogradov–Vaughan, Jackson, Rosser–Schoenfeld, the prime number theorem, von Koch, Bringmann–Wellnitz, [BR, Theorem 22.1], and for Proposition 3 the GRH bound for π ( x , χ ) \pi(x,\chi) π ( x , χ )
The scripts, the pseudocode of the solver, the benchmark files and the Lean project are in the public repository arcs-algo, github.com/anthem87/arcs-algo (AGPL-3.0-or-later; the version described here is commit 372ba68). This paper is archived at doi:10.5281/zenodo.23912385 (all versions).
AI tools were used to assist with computations, the drafting of text and the checking of proofs. All results were verified by the author, who takes full responsibility for the content.
