Introduction
For a finite set A ⊂ R A \subset\mathbb{R} A ⊂ R Sidon sets and Golomb rulers , in which all positive differences are distinct [17 , 18 , 23 , 42 , 45 ] . At the other end are arithmetic progressions, whose difference set A − A A-A A − A ∣ A − A ∣ = 2 ∣ A ∣ − 1 |A-A|=2|A|-1 ∣ A − A ∣ = 2∣ A ∣ − 1 [41 , 52 ] . Between them lie the ( v , k , λ ) (v,k,\lambda) ( v , k , λ ) [6 , 9 , 47 ] .
In this paper we impose a regularity of a different kind: we prescribe the shape of the set of distinct differences and ask that it be a geometric progression. The simplest non-trivial case already singles out the golden ratio φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2 φ = ( 1 + 5 ) /2 u > 0 u>0 u > 0
a = φ u , b = φ 2 u , F > a + b , S = ( F , F − a , F − b , F − a − b ) . a=\varphi u,\qquad b=\varphi^{2}u,\qquad F>a+b,\qquad S=(F,F-a,F-b,F-a-b). a = φ u , b = φ 2 u , F > a + b , S = ( F , F − a , F − b , F − a − b ) . We call S S S golden quadruple . Because φ 2 = φ + 1 \varphi^{2}=\varphi+1 φ 2 = φ + 1 b − a = u b-a=u b − a = u a + b = φ 3 u a+b=\varphi^{3}u a + b = φ 3 u
The configuration itself is classical. As a set, S S S u { 0 , φ , φ 2 , φ 3 } u\{0,\varphi,\varphi^{2},\varphi^{3}\} u { 0 , φ , φ 2 , φ 3 } { 0 , φ , φ 2 , φ 3 } \{0,\varphi,\varphi^{2},\varphi^{3}\} { 0 , φ , φ 2 , φ 3 } Elements XIII.8, two diagonals of a regular pentagon that subtend consecutive angles cut one another in extreme and mean ratio, and their greater segments are equal to the side of the pentagon [19 ] . The details are recalled in Remark 3.2. Up to dilation it is also the sequence 0 , 1 , φ , φ 2 0,1,\varphi,\varphi^{2} 0 , 1 , φ , φ 2 [30 (3.9)] from [30 (2.4)(b)] . Neither the set nor its difference multiset (Theorem 1) is new, and neither is the bound of four points (Remark 4.5); what this paper adds are the rigidity theorem, an explicit geometric formulation of uniqueness with an elementary proof, and the analysis of the cosine sum.
Main results. Write D = { u , φ u , φ 2 u , φ 3 u } \mathcal{D}=\{u,\varphi u,\varphi^{2}u,\varphi^{3}u\} D = { u , φ u , φ 2 u , φ 3 u }
(1) Theorem 1 (classical). The six pairwise differences of S S S u , φ u , φ 2 u , φ 3 u u,\varphi u,\varphi^{2}u,\varphi^{3}u u , φ u , φ 2 u , φ 3 u 1 , 2 , 2 , 1 1,2,2,1 1 , 2 , 2 , 1
(2) Theorem 2 and Theorem 2′. Every four-element set with all its positive differences in D \mathcal{D} D { c φ n : n ∈ Z } \{c\varphi^{n}:n\in\mathbb{Z}\} { c φ n : n ∈ Z } c > 0 c>0 c > 0 [30 ] (Remark 4.5), and we include a self-contained elementary proof.
(3) Theorem 3. For a base x > 1 x>1 x > 1 x ≠ φ x\ne\varphi x = φ ( F , F − x u , F − x 2 u , F − x u − x 2 u ) (F,F-xu,F-x^{2}u,F-xu-x^{2}u) ( F , F − xu , F − x 2 u , F − xu − x 2 u ) u , x u , x 2 u , x 3 u u,xu,x^{2}u,x^{3}u u , xu , x 2 u , x 3 u ρ − ( x ) = ( x 2 − x − 1 ) u \rho_{-}(x)=(x^{2}-x-1)u ρ − ( x ) = ( x 2 − x − 1 ) u ρ + ( x ) = x ( x 2 − x − 1 ) u \rho_{+}(x)=x(x^{2}-x-1)u ρ + ( x ) = x ( x 2 − x − 1 ) u
(4) Theorem 3′. If x > 1 x>1 x > 1 T ⊂ R T\subset\mathbb{R} T ⊂ R Δ ( T ) = { c , c x , c x 2 , c x 3 } \Delta(T)=\{c,cx,cx^{2},cx^{3}\} Δ ( T ) = { c , c x , c x 2 , c x 3 } x = φ x=\varphi x = φ T T T c { 0 , φ , φ 2 , φ 3 } c\{0,\varphi,\varphi^{2},\varphi^{3}\} c { 0 , φ , φ 2 , φ 3 }
(5) Theorem 4. The cosine sum
C S ( t ) = cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t ) CS(t)=\cos(2\pi ut)+2\cos(2\pi\varphi ut)+2\cos(2\pi\varphi^{2}ut)+\cos(2\pi\varphi^{3}ut) C S ( t ) = cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t ) is not periodic but is almost periodic in the sense of Bohr [10 ] . Moreover,
C S ( t ) = 8 cos 2 ( π a t ) cos 2 ( π b t ) − 2 , CS(t)=8\cos^{2}(\pi at)\cos^{2}(\pi bt)-2, C S ( t ) = 8 cos 2 ( π a t ) cos 2 ( π b t ) − 2 , and the numbers F i b n / u \mathrm{Fib}_{n}/u Fib n / u sup t ∣ C S ( t + F i b n / u ) − C S ( t ) ∣ = 6 3 π φ − n − 2 3 π φ − 3 n + O ( φ − 5 n ) \sup_{t}|CS(t+\mathrm{Fib}_{n}/u)-CS(t)|=6\sqrt{3}\pi\varphi^{-n}-2\sqrt{3}\pi\varphi^{-3n}+O(\varphi^{-5n}) sup t ∣ C S ( t + Fib n / u ) − C S ( t ) ∣ = 6 3 π φ − n − 2 3 π φ − 3 n + O ( φ − 5 n ) L ( ε ) ≍ 1 / ε L(\varepsilon)\asymp1/\varepsilon L ( ε ) ≍ 1/ ε
Scale. All statements are invariant under dilations, so the only role of a scale is to fix a normalisation. For an arbitrary K > 0 K>0 K > 0 golden ladder ν n = K φ − n \nu_{n}=K\varphi^{-n} ν n = K φ − n n ∈ Z n\in\mathbb{Z} n ∈ Z ν n = ν n + 1 + ν n + 2 \nu_{n}=\nu_{n+1}+\nu_{n+2} ν n = ν n + 1 + ν n + 2 u = ν 203 u=\nu_{203} u = ν 203 a = ν 202 a=\nu_{202} a = ν 202 b = ν 201 b=\nu_{201} b = ν 201 a + b = ν 200 a+b=\nu_{200} a + b = ν 200 S S S { ν 200 , ν 201 , ν 202 , ν 203 } \{\nu_{200},\nu_{201},\nu_{202},\nu_{203}\} { ν 200 , ν 201 , ν 202 , ν 203 } K = c 5 / ( G h ) K=\sqrt{c^{5}/(Gh)} K = c 5 / ( G h )
What is classical and what is new. Everything here is elementary. The set { 0 , φ , φ 2 , φ 3 } \{0,\varphi,\varphi^{2},\varphi^{3}\} { 0 , φ , φ 2 , φ 3 } Elements XIII.8), and Theorem 1 is an immediate consequence of φ 2 = φ + 1 \varphi^{2}=\varphi+1 φ 2 = φ + 1 § 8 ).
Theorems 2(ii) and 2′(ii) (uniqueness; not claimed as new): an explicit geometric formulation with our own elementary proof; they can also be derived from Nagell’s list of exceptional units via normalisation [31 , 37 , 38 , 39 ] , see Remark 4.3. The four-point configuration is unique up to translation (and, in Theorem 2′(ii), dilation by a power of φ \varphi φ
New: Theorem 3′ (rigidity): among all ratios x > 1 x > 1 x > 1 x = φ x = \varphi x = φ { c , c x , c x 2 , c x 3 } \{c, cx, cx^2, cx^3\} { c , c x , c x 2 , c x 3 }
New: the quantitative analysis of the cosine sum C S CS C S F i b n / u \mathrm{Fib}_n/u Fib n / u Proposition 6.8 ), and the order L ( ε ) ≍ 1 / ε L(\varepsilon) \asymp1/\varepsilon L ( ε ) ≍ 1/ ε Proposition 6.2 ) and the almost periods F i b n / u \mathrm{Fib}_n/u Fib n / u Proposition 6.5 ), which are elementary; the explicit value C S ( F i b n / u ) CS(\mathrm{Fib}_n/u) C S ( Fib n / u ) Proposition 6.5 (iii) is ours. Theorem 4 (non-periodicity and Bohr almost periodicity) is a direct consequence of classical facts and is included for completeness.
The bound of four points (Theorems 2 (i) and 2 ′(iii)), i.e. N ( φ ) = 4 N(\varphi) = 4 N ( φ ) = 4 Definition 4.4 , is not new: it follows from Lenstra’s work on Euclidean number fields [30 (2.2), (3.9)] , see Remark 4.5. Version 1 of this preprint listed it among the new results; we correct this here. Our proof is elementary and independent of [30 ] . Likewise Lemma 4.2 and Theorem 2 ′(i) follow from Nagell’s determination of the exceptional units of Q ( 5 ) \mathbb{Q}(\sqrt{5}) Q ( 5 ) [37 , 38 , 39 ] . Theorem 3 and Proposition 5.3 are elementary companions of Theorem 3′ for quadruples of the special form S x S_x S x § 8 discusses the relation to the literature in detail; it also shows that some of these phenomena are genuinely specific to φ \varphi φ
Organisation. Section 2 fixes notation. Sections 3–6 contain the main theorems and their strengthenings. Section 7 gives numerical examples and tables, Section 8 discusses the literature and open questions, and Section 9 describes which of the results have been formally verified in Lean 4.
Preliminaries
Throughout, N = { 0 , 1 , 2 , … } \mathbb{N} = \{0, 1, 2, \ldots\} N = { 0 , 1 , 2 , … } ∥ θ ∥ = min m ∈ Z ∣ θ − m ∣ \lVert\theta\rVert= \min_{m \in\mathbb{Z}} \lvert\theta-m\rvert ∥ θ ∥ = min m ∈ Z ∣ θ − m ∣ θ ∈ R \theta\in\mathbb{R} θ ∈ R e ( θ ) = e 2 π i θ e(\theta) = e^{2\pi i\theta} e ( θ ) = e 2 π i θ F i b 0 = 0 \mathrm{Fib}_0 = 0 Fib 0 = 0 F i b 1 = 1 \mathrm{Fib}_1 = 1 Fib 1 = 1 F i b n + 1 = F i b n + F i b n − 1 \mathrm{Fib}_{n+1} = \mathrm{Fib}_n + \mathrm{Fib}_{n-1} Fib n + 1 = Fib n + Fib n − 1
Differences
Definition 2.1. Let A ⊂ R A \subset\mathbb{R} A ⊂ R representation function of A A A
r A ( d ) = # { ( x , y ) ∈ A × A : x − y = d } ( d > 0 ) , r_A(d) = \#\{(x,y) \in A \times A : x-y = d\} \qquad(d > 0), r A ( d ) = # {( x , y ) ∈ A × A : x − y = d } ( d > 0 ) , the positive difference set is Δ ( A ) = { d > 0 : r A ( d ) ≥ 1 } \Delta(A) = \{d > 0 : r_A(d) \ge1\} Δ ( A ) = { d > 0 : r A ( d ) ≥ 1 } difference multiset of A A A d ∈ Δ ( A ) d \in\Delta(A) d ∈ Δ ( A ) r A ( d ) r_A(d) r A ( d ) ( ∣ A ∣ 2 ) \binom{|A|}{2} ( 2 ∣ A ∣ ) A − A = { x − y : x , y ∈ A } A-A = \{x-y : x,y \in A\} A − A = { x − y : x , y ∈ A }
Definition 2.2 (classical notions, used in Section 8). A finite A ⊂ R A \subset\mathbb{R} A ⊂ R Sidon set if r A ( d ) ≤ 1 r_A(d) \le1 r A ( d ) ≤ 1 Golomb ruler is a Sidon subset of Z \mathbb{Z} Z perfect if Δ ( A ) = { 1 , 2 , … , max A − min A } \Delta(A) = \{1,2,\ldots,\max A-\min A\} Δ ( A ) = { 1 , 2 , … , max A − min A } G G G v v v k k k B ⊆ G B \subseteq G B ⊆ G ( v , k , λ ) (v,k,\lambda) ( v , k , λ ) difference set if every g ∈ G ∖ { 0 } g \in G\setminus\{0\} g ∈ G ∖ { 0 } λ \lambda λ g = x − y g=x-y g = x − y x , y ∈ B x,y \in B x , y ∈ B
Lemma 2.3. For θ , θ ′ ∈ R \theta,\theta' \in\mathbb{R} θ , θ ′ ∈ R m ∈ Z m \in\mathbb{Z} m ∈ Z ∥ θ + m ∥ = ∥ θ ∥ = ∥ − θ ∥ \|\theta+m\|=\|\theta\|=\|-\theta\| ∥ θ + m ∥ = ∥ θ ∥ = ∥ − θ ∥ ∥ θ + θ ′ ∥ ≤ ∥ θ ∥ + ∥ θ ′ ∥ \|\theta+\theta'\|\le\|\theta\|+\|\theta'\| ∥ θ + θ ′ ∥ ≤ ∥ θ ∥ + ∥ θ ′ ∥ ∣ e ( θ ) − 1 ∣ = 2 ∣ sin π θ ∣ ≤ 2 π ∥ θ ∥ |e(\theta)-1|=2|\sin\pi\theta|\le2\pi\|\theta\| ∣ e ( θ ) − 1∣ = 2∣ sin π θ ∣ ≤ 2 π ∥ θ ∥
Proof. The first two identities are clear. If m , m ′ ∈ Z m,m' \in\mathbb{Z} m , m ′ ∈ Z θ , θ ′ \theta,\theta' θ , θ ′ ∥ θ + θ ′ ∥ ≤ ∣ θ + θ ′ − m − m ′ ∣ ≤ ∥ θ ∥ + ∥ θ ′ ∥ \|\theta+\theta'\|\le|\theta+\theta'-m-m'|\le\|\theta\|+\|\theta'\| ∥ θ + θ ′ ∥ ≤ ∣ θ + θ ′ − m − m ′ ∣ ≤ ∥ θ ∥ + ∥ θ ′ ∥ ∣ e ( θ ) − 1 ∣ = ∣ e π i θ − e − π i θ ∣ = 2 ∣ sin π θ ∣ |e(\theta)-1|=|e^{\pi i\theta}-e^{-\pi i\theta}|=2|\sin\pi\theta| ∣ e ( θ ) − 1∣ = ∣ e π i θ − e − π i θ ∣ = 2∣ sin π θ ∣ m m m θ \theta θ ∣ sin π θ ∣ = ∣ sin π ( θ − m ) ∣ ≤ π ∣ θ − m ∣ = π ∥ θ ∥ |\sin\pi\theta|=|\sin\pi(\theta-m)|\le\pi|\theta-m|=\pi\|\theta\| ∣ sin π θ ∣ = ∣ sin π ( θ − m ) ∣ ≤ π ∣ θ − m ∣ = π ∥ θ ∥
The golden ratio
Let φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2 φ = ( 1 + 5 ) /2 φ ˉ = ( 1 − 5 ) / 2 \bar{\varphi}=(1-\sqrt{5})/2 φ ˉ = ( 1 − 5 ) /2
Lemma 2.4. (i) φ 2 = φ + 1 \varphi^2=\varphi+1 φ 2 = φ + 1 φ + φ ˉ = 1 \varphi+\bar{\varphi}=1 φ + φ ˉ = 1 φ φ ˉ = − 1 \varphi\bar{\varphi}=-1 φ φ ˉ = − 1 φ ˉ = − φ − 1 \bar{\varphi}=-\varphi^{-1} φ ˉ = − φ − 1
(ii) φ n + 2 = φ n + 1 + φ n \varphi^{n+2}=\varphi^{n+1}+\varphi^n φ n + 2 = φ n + 1 + φ n n ∈ Z n\in\mathbb{Z} n ∈ Z φ 3 = 2 φ + 1 \varphi^3=2\varphi+1 φ 3 = 2 φ + 1 φ 4 = 3 φ + 2 \varphi^4=3\varphi+2 φ 4 = 3 φ + 2 φ − 1 = φ − 1 \varphi-1=\varphi^{-1} φ − 1 = φ − 1 2 − φ = φ − 2 2-\varphi=\varphi^{-2} 2 − φ = φ − 2
(iii) 3 / 2 < φ < 2 3/2<\varphi<2 3/2 < φ < 2 φ 2 < 3 \varphi^2<3 φ 2 < 3 φ 3 > 4 \varphi^3>4 φ 3 > 4
(iv) φ \varphi φ 1 1 1 φ \varphi φ Q \mathbb{Q} Q
(v) For x > 1 x>1 x > 1 x 2 − x − 1 = ( x − φ ) ( x − φ ˉ ) x^2-x-1=(x-\varphi)(x-\bar{\varphi}) x 2 − x − 1 = ( x − φ ) ( x − φ ˉ ) x − φ ˉ > 0 x-\bar{\varphi}>0 x − φ ˉ > 0 x 2 − x − 1 = 0 x^2-x-1=0 x 2 − x − 1 = 0 x = φ x=\varphi x = φ sgn ( x 2 − x − 1 ) = sgn ( x − φ ) \operatorname{sgn}(x^2-x-1)=\operatorname{sgn}(x-\varphi) sgn ( x 2 − x − 1 ) = sgn ( x − φ )
(vi) The map n ↦ φ n n\mapsto\varphi^n n ↦ φ n Z \mathbb{Z} Z
Proof. (i) φ 2 = ( 6 + 2 5 ) / 4 = ( 3 + 5 ) / 2 = φ + 1 \varphi^2=(6+2\sqrt{5})/4=(3+\sqrt{5})/2=\varphi+1 φ 2 = ( 6 + 2 5 ) /4 = ( 3 + 5 ) /2 = φ + 1 φ ˉ = − 1 / φ \bar{\varphi}=-1/\varphi φ ˉ = − 1/ φ φ φ ˉ = − 1 \varphi\bar{\varphi}=-1 φ φ ˉ = − 1 φ 2 = φ + 1 \varphi^2=\varphi+1 φ 2 = φ + 1 φ n \varphi^n φ n n = 1 , 2 n=1,2 n = 1 , 2 φ 3 = φ 2 + φ = 2 φ + 1 \varphi^3=\varphi^2+\varphi=2\varphi+1 φ 3 = φ 2 + φ = 2 φ + 1 φ 4 = φ 3 + φ 2 = 3 φ + 2 \varphi^4=\varphi^3+\varphi^2=3\varphi+2 φ 4 = φ 3 + φ 2 = 3 φ + 2 n = − 1 n=-1 n = − 1 φ = 1 + φ − 1 \varphi=1+\varphi^{-1} φ = 1 + φ − 1 n = − 2 n=-2 n = − 2 1 = φ − 1 + φ − 2 1=\varphi^{-1}+\varphi^{-2} 1 = φ − 1 + φ − 2 φ − 2 = 1 − ( φ − 1 ) = 2 − φ \varphi^{-2}=1-(\varphi-1)=2-\varphi φ − 2 = 1 − ( φ − 1 ) = 2 − φ 2 < 5 < 3 2<\sqrt{5}<3 2 < 5 < 3 φ 2 = φ + 1 < 3 \varphi^2=\varphi+1<3 φ 2 = φ + 1 < 3 φ 3 = 2 φ + 1 > 4 \varphi^3=2\varphi+1>4 φ 3 = 2 φ + 1 > 4 φ = p / q ∈ Q \varphi=p/q\in\mathbb{Q} φ = p / q ∈ Q 5 = 2 p / q − 1 ∈ Q \sqrt{5}=2p/q-1\in\mathbb{Q} 5 = 2 p / q − 1 ∈ Q 5 = m / n \sqrt{5}=m/n 5 = m / n m 2 = 5 n 2 m^2=5n^2 m 2 = 5 n 2 5 ∣ m 5\mid m 5 ∣ m 25 ∣ 5 n 2 25\mid5n^2 25 ∣ 5 n 2 5 ∣ n 2 5\mid n^2 5 ∣ n 2 5 ∣ n 5\mid n 5 ∣ n r + s φ = 0 r+s\varphi=0 r + s φ = 0 r , s ∈ Q r,s\in\mathbb{Q} r , s ∈ Q s ≠ 0 s\ne0 s = 0 φ = − r / s ∈ Q \varphi=-r/s\in\mathbb{Q} φ = − r / s ∈ Q x 2 − x − 1 x^2-x-1 x 2 − x − 1 φ \varphi φ φ ˉ < 0 < 1 < x \bar{\varphi}<0<1<x φ ˉ < 0 < 1 < x φ > 1 \varphi>1 φ > 1
The construction
Definition 2.5 (golden quadruple). Fix u > 0 u>0 u > 0 a = φ u a=\varphi u a = φ u b = φ 2 u b=\varphi^2u b = φ 2 u F > a + b F>a+b F > a + b golden quadruple is
S = S ( F , u ) = ( s 1 , s 2 , s 3 , s 4 ) = ( F , F − a , F − b , F − a − b ) . S=S(F,u)=(s_1,s_2,s_3,s_4)=(F,F-a,F-b,F-a-b). S = S ( F , u ) = ( s 1 , s 2 , s 3 , s 4 ) = ( F , F − a , F − b , F − a − b ) . We write D = D ( u ) = { u , φ u , φ 2 u , φ 3 u } D=D(u)=\{u,\varphi u,\varphi^2u,\varphi^3u\} D = D ( u ) = { u , φ u , φ 2 u , φ 3 u } 0 < a < b < a + b < F 0<a<b<a+b<F 0 < a < b < a + b < F s 1 > s 2 > s 3 > s 4 > 0 s_1>s_2>s_3>s_4>0 s 1 > s 2 > s 3 > s 4 > 0 F > a + b F>a+b F > a + b S ⊂ ( 0 , ∞ ) S\subset(0,\infty) S ⊂ ( 0 , ∞ )
The name should not be confused with the golden quadruples of Weiß [57 Theorem 3] : four collinear points characterised projectively by their cross ratios. That notion is projectively invariant and covers a wider class than the metric condition of Definition 2.5.
Definition 2.6 (golden ladder). For K > 0 K > 0 K > 0 n ∈ Z n \in\mathbb{Z} n ∈ Z ν n = ν n ( K ) = K φ − n \nu_n = \nu_n(K) = K\varphi^{-n} ν n = ν n ( K ) = K φ − n
Lemma 2.7. For every n ∈ Z n \in\mathbb{Z} n ∈ Z ν n = ν n + 1 + ν n + 2 \nu_n = \nu_{n+1} + \nu_{n+2} ν n = ν n + 1 + ν n + 2 ν n = φ ν n + 1 \nu_n = \varphi\nu_{n+1} ν n = φ ν n + 1 u = ν 203 u = \nu_{203} u = ν 203 a = ν 202 a = \nu_{202} a = ν 202 b = ν 201 b = \nu_{201} b = ν 201 a + b = ν 200 a + b = \nu_{200} a + b = ν 200 b − a = ν 203 b - a = \nu_{203} b − a = ν 203
Proof. ν n − ν n + 1 − ν n + 2 = K φ − n − 2 ( φ 2 − φ − 1 ) = 0 \nu_n-\nu_{n+1}-\nu_{n+2}=K\varphi^{-n-2}(\varphi^2-\varphi-1)=0 ν n − ν n + 1 − ν n + 2 = K φ − n − 2 ( φ 2 − φ − 1 ) = 0 ν n / ν n + 1 = φ \nu_n/\nu_{n+1}=\varphi ν n / ν n + 1 = φ a = φ ν 203 = ν 202 a=\varphi\nu_{203}=\nu_{202} a = φ ν 203 = ν 202 b = φ 2 ν 203 = ν 201 b=\varphi^2\nu_{203}=\nu_{201} b = φ 2 ν 203 = ν 201 n = 200 n=200 n = 200 a + b = ν 202 + ν 201 = ν 200 a+b=\nu_{202}+\nu_{201}=\nu_{200} a + b = ν 202 + ν 201 = ν 200 n = 201 n=201 n = 201 b − a = ν 201 − ν 202 = ν 203 b-a=\nu_{201}-\nu_{202}=\nu_{203} b − a = ν 201 − ν 202 = ν 203
Example 2.8 (the Planck scale). Let K P = c 5 / ( G h ) K_{\mathrm{P}}=\sqrt{c^5/(Gh)} K P = c 5 / ( G h ) c = 299 792 458 c=299\,792\,458 c = 299 792 458 G = 6.674 30 × 10 − 11 G=6.674\,30 \times10^{-11} G = 6.674 30 × 1 0 − 11 h = 6.626 070 15 × 10 − 34 h=6.626\,070\,15 \times10^{-34} h = 6.626 070 15 × 1 0 − 34 [54 ] ; c c c h h h G G G 2.2 × 10 − 5 2.2 \times10^{-5} 2.2 × 1 0 − 5
K P = 7.399 82 ( 8 ) × 10 42 , K_{\mathrm{P}}=7.399\,82(8)\times10^{42}, K P = 7.399 82 ( 8 ) × 1 0 42 , the relative uncertainty 1.1 × 10 − 5 1.1\times10^{-5} 1.1 × 1 0 − 5 G G G φ 203 ≈ 2.657 61 × 10 42 \varphi^{203}\approx2.657\,61\times10^{42} φ 203 ≈ 2.657 61 × 1 0 42 ν 203 ( K P ) ≈ 2.784 39 \nu_{203}(K_{\mathrm{P}})\approx2.784\,39 ν 203 ( K P ) ≈ 2.784 39 h h h ℏ = h / ( 2 π ) \hbar=h/(2\pi) ℏ = h / ( 2 π ) 2 π K P ≈ 1.854 86 × 10 43 \sqrt{2\pi}K_{\mathrm{P}}\approx1.854\,86\times10^{43} 2 π K P ≈ 1.854 86 × 1 0 43 K P K_{\mathrm{P}} K P
Almost periodic functions and cosine sums
Definition 2.9 (Bohr [8 , 10 , 11 , 14 ] ). A set E ⊆ R E\subseteq\mathbb{R} E ⊆ R relatively dense if there is L > 0 L>0 L > 0 L L L E E E f : R → C f:\mathbb{R}\to\mathbb{C} f : R → C ε > 0 \varepsilon>0 ε > 0 ε \varepsilon ε almost periods is
E ( f ; ε ) = { τ ∈ R : sup t ∈ R ∣ f ( t + τ ) − f ( t ) ∣ ≤ ε } . E(f;\varepsilon)=\left\{\tau\in\mathbb{R}:\sup_{t\in\mathbb{R}}|f(t+\tau)-f(t)|\leq\varepsilon\right\}. E ( f ; ε ) = { τ ∈ R : t ∈ R sup ∣ f ( t + τ ) − f ( t ) ∣ ≤ ε } . The function f f f almost periodic (in the sense of Bohr) if E ( f ; ε ) E(f;\varepsilon) E ( f ; ε ) ε > 0 \varepsilon>0 ε > 0 periodic if f ( t + T ) = f ( t ) f(t+T)=f(t) f ( t + T ) = f ( t ) t t t T > 0 T>0 T > 0
A continuous periodic function with period T T T E ( f ; ε ) ⊇ T Z E(f;\varepsilon)\supseteq T\mathbb{Z} E ( f ; ε ) ⊇ T Z T T T T Z T\mathbb{Z} T Z
Definition 2.10 (cosine sum). For a finite A ⊂ R A\subset\mathbb{R} A ⊂ R
C A ( t ) = ∑ x , y ∈ A x > y cos ( 2 π ( x − y ) t ) = ∑ d ∈ Δ ( A ) r A ( d ) cos ( 2 π d t ) , A ^ ( t ) = ∑ x ∈ A e ( x t ) . C_A(t)=\sum_{\substack{x,y\in A\\x>y}}\cos(2\pi(x-y)t)=\sum_{d\in\Delta(A)}r_A(d)\cos(2\pi dt),\qquad\widehat{A}(t)=\sum_{x\in A}e(xt). C A ( t ) = x , y ∈ A x > y ∑ cos ( 2 π ( x − y ) t ) = d ∈ Δ ( A ) ∑ r A ( d ) cos ( 2 π d t ) , A ( t ) = x ∈ A ∑ e ( x t ) . The difference multiset of the golden quadruple
Lemma 3.1 (two-dimensional cubes). Let 0 < α < β 0<\alpha<\beta 0 < α < β Q = { 0 , α , β , α + β } = { 0 , α } + { 0 , β } Q=\{0,\alpha,\beta,\alpha+\beta\}=\{0,\alpha\}+\{0,\beta\} Q = { 0 , α , β , α + β } = { 0 , α } + { 0 , β } Q Q Q
α − 0 = α , β − 0 = β , ( α + β ) − 0 = α + β , β − α , ( α + β ) − α = β , ( α + β ) − β = α , \alpha-0=\alpha,\quad\beta-0=\beta,\quad(\alpha+\beta)-0=\alpha+\beta,\quad\beta-\alpha,\quad(\alpha+\beta)-\alpha=\beta,\quad(\alpha+\beta)-\beta=\alpha, α − 0 = α , β − 0 = β , ( α + β ) − 0 = α + β , β − α , ( α + β ) − α = β , ( α + β ) − β = α , so the difference multiset of Q Q Q { β − α , α , α , β , β , α + β } \{\beta-\alpha,\alpha,\alpha,\beta,\beta,\alpha+\beta\} { β − α , α , α , β , β , α + β } F ∈ R F\in\mathbb{R} F ∈ R F − Q F-Q F − Q t + Q t+Q t + Q Q Q Q ( α + β ) − Q = Q (\alpha+\beta)-Q=Q ( α + β ) − Q = Q
Proof. The list is a direct enumeration of the ( 4 2 ) = 6 \binom{4}{2}=6 ( 2 4 ) = 6 x > y x>y x > y Q Q Q ( F − y ) − ( F − x ) = x − y (F-y)-(F-x)=x-y ( F − y ) − ( F − x ) = x − y ( t + x ) − ( t + y ) = x − y (t+x)-(t+y)=x-y ( t + x ) − ( t + y ) = x − y { α + β , β , α , 0 } = Q \{\alpha+\beta,\beta,\alpha,0\}=Q { α + β , β , α , 0 } = Q
Theorem 1. The six pairwise differences of the golden quadruple are
s 1 − s 2 = φ u , s 1 − s 3 = φ 2 u , s 1 − s 4 = φ 3 u , s_1-s_2=\varphi u,\qquad s_1-s_3=\varphi^2u,\qquad s_1-s_4=\varphi^3u, s 1 − s 2 = φ u , s 1 − s 3 = φ 2 u , s 1 − s 4 = φ 3 u , s 2 − s 3 = u , s 2 − s 4 = φ 2 u , s 3 − s 4 = φ u . s_2-s_3=u,\qquad s_2-s_4=\varphi^2u,\qquad s_3-s_4=\varphi u. s 2 − s 3 = u , s 2 − s 4 = φ 2 u , s 3 − s 4 = φ u . Consequently Δ ( S ) = D = { u , φ u , φ 2 u , φ 3 u } \Delta(S)=D=\{u,\varphi u,\varphi^2u,\varphi^3u\} Δ ( S ) = D = { u , φ u , φ 2 u , φ 3 u } φ \varphi φ
r S ( u ) = 1 , r S ( φ u ) = 2 , r S ( φ 2 u ) = 2 , r S ( φ 3 u ) = 1. r_S(u)=1,\qquad r_S(\varphi u)=2,\qquad r_S(\varphi^2u)=2,\qquad r_S(\varphi^3u)=1. r S ( u ) = 1 , r S ( φ u ) = 2 , r S ( φ 2 u ) = 2 , r S ( φ 3 u ) = 1. Proof. Directly: s 1 − s 2 = a = φ u s_1-s_2=a=\varphi u s 1 − s 2 = a = φ u s 1 − s 3 = b = φ 2 u s_1-s_3=b=\varphi^2u s 1 − s 3 = b = φ 2 u s 2 − s 4 = b s_2-s_4=b s 2 − s 4 = b s 3 − s 4 = a s_3-s_4=a s 3 − s 4 = a s 1 − s 4 = a + b = ( φ + φ 2 ) u = φ ( 1 + φ ) u = φ ⋅ φ 2 u = φ 3 u s_1-s_4=a+b=(\varphi+\varphi^2)u=\varphi(1+\varphi)u=\varphi\cdot\varphi^2u=\varphi^3u s 1 − s 4 = a + b = ( φ + φ 2 ) u = φ ( 1 + φ ) u = φ ⋅ φ 2 u = φ 3 u s 2 − s 3 = b − a = ( φ 2 − φ ) u = u s_2-s_3=b-a=(\varphi^2-\varphi)u=u s 2 − s 3 = b − a = ( φ 2 − φ ) u = u φ 2 = φ + 1 \varphi^2=\varphi+1 φ 2 = φ + 1 φ > 1 \varphi>1 φ > 1 u < φ u < φ 2 u < φ 3 u u<\varphi u<\varphi^2u<\varphi^3u u < φ u < φ 2 u < φ 3 u S = F − Q S=F-Q S = F − Q Q = { 0 , a } + { 0 , b } Q=\{0,a\}+\{0,b\} Q = { 0 , a } + { 0 , b } β − α = u \beta-\alpha=u β − α = u α + β = φ 2 u \alpha+\beta=\varphi^2u α + β = φ 2 u
The set u { 0 , φ , φ 2 , φ 3 } u\{0,\varphi,\varphi^2,\varphi^3\} u { 0 , φ , φ 2 , φ 3 } S S S A B C D E ABCDE A B C D E φ 3 u \varphi^3u φ 3 u A C AC A C B E BE B E H H H B D BD B D H ′ H' H ′ A C AC A C B E BE B E B B B A A A Elements XIII.8 [19 Book XIII, Prop. 8] the point H H H A C AC A C C H CH C H φ 3 u / φ = φ 2 u \varphi^3u/\varphi=\varphi^2u φ 3 u / φ = φ 2 u A H = φ 3 u − φ 2 u = φ u AH=\varphi^3u-\varphi^2u=\varphi u A H = φ 3 u − φ 2 u = φ u A C AC A C A A A C C C B E BE B E B D BD B D H H H H ′ H' H ′ C H ′ = A H = φ u CH'=AH=\varphi u C H ′ = A H = φ u A H ′ = φ 2 u AH'=\varphi^2u A H ′ = φ 2 u A A A A , H , H ′ , C A,H,H',C A , H , H ′ , C A C AC A C A C E B D ACEBD A C E B D 0 , φ u , φ 2 u , φ 3 u 0,\varphi u,\varphi^2u,\varphi^3u 0 , φ u , φ 2 u , φ 3 u
Corollary 3.3 (golden ladder). Let K > 0 K>0 K > 0 u = ν 203 u=\nu_{203} u = ν 203 F > ν 200 F>\nu_{200} F > ν 200
Δ ( S ) = { ν 200 , ν 201 , ν 202 , ν 203 } , r S ( ν 200 ) = r S ( ν 203 ) = 1 , r S ( ν 201 ) = r S ( ν 202 ) = 2. \Delta(S)=\{\nu_{200},\nu_{201},\nu_{202},\nu_{203}\},\qquad r_S(\nu_{200})=r_S(\nu_{203})=1,\qquad r_S(\nu_{201})=r_S(\nu_{202})=2. Δ ( S ) = { ν 200 , ν 201 , ν 202 , ν 203 } , r S ( ν 200 ) = r S ( ν 203 ) = 1 , r S ( ν 201 ) = r S ( ν 202 ) = 2. More generally, for m ∈ Z m\in\mathbb{Z} m ∈ Z u = ν m + 3 u=\nu_{m+3} u = ν m + 3 F > ν m F>\nu_m F > ν m Δ ( S ) = { ν m , ν m + 1 , ν m + 2 , ν m + 3 } \Delta(S)=\{\nu_m,\nu_{m+1},\nu_{m+2},\nu_{m+3}\} Δ ( S ) = { ν m , ν m + 1 , ν m + 2 , ν m + 3 }
Proof. By Lemma 2.7, φ k ν m + 3 = ν m + 3 − k \varphi^k\nu_{m+3}=\nu_{m+3-k} φ k ν m + 3 = ν m + 3 − k D ( ν m + 3 ) = { ν m + 3 , ν m + 2 , ν m + 1 , ν m } D(\nu_{m+3})=\{\nu_{m+3},\nu_{m+2},\nu_{m+1},\nu_m\} D ( ν m + 3 ) = { ν m + 3 , ν m + 2 , ν m + 1 , ν m } m = 200 m=200 m = 200 a + b = ν 200 a+b=\nu_{200} a + b = ν 200 F > ν 200 F>\nu_{200} F > ν 200 F > a + b F>a+b F > a + b
(a) As a set, S = ( F − a − b ) + ( { 0 , a } + { 0 , b } ) = ( F − φ 3 u ) + u { 0 , φ , φ 2 , φ 3 } S=(F-a-b)+(\{0,a\}+\{0,b\})=(F-\varphi^3u)+u\{0,\varphi,\varphi^2,\varphi^3\} S = ( F − a − b ) + ({ 0 , a } + { 0 , b }) = ( F − φ 3 u ) + u { 0 , φ , φ 2 , φ 3 } φ + φ 2 = φ 3 \varphi+\varphi^2=\varphi^3 φ + φ 2 = φ 3 S S S u ( { 0 , φ } + { 0 , φ 2 } ) u(\{0,\varphi\}+\{0,\varphi^2\}) u ({ 0 , φ } + { 0 , φ 2 }) β ≠ 2 α \beta\ne2\alpha β = 2 α { 1 , 1 , 2 , 2 } \{1,1,2,2\} { 1 , 1 , 2 , 2 } 1 , 2 , 2 , 1 1,2,2,1 1 , 2 , 2 , 1 α < β < 2 α \alpha<\beta<2\alpha α < β < 2 α β − α < α < β < α + β \beta-\alpha<\alpha<\beta<\alpha+\beta β − α < α < β < α + β 2 , 1 , 2 , 1 2,1,2,1 2 , 1 , 2 , 1 β > 2 α \beta>2\alpha β > 2 α α < β − α < β < α + β \alpha<\beta-\alpha<\beta<\alpha+\beta α < β − α < β < α + β β / α = φ ∈ ( 1 , 2 ) \beta/\alpha=\varphi\in(1,2) β / α = φ ∈ ( 1 , 2 ) 1 , 2 , 2 , 1 1,2,2,1 1 , 2 , 2 , 1
(b) Every three-element subset of S S S φ \varphi φ { s 1 , s 2 , s 3 } \{s_1,s_2,s_3\} { s 1 , s 2 , s 3 } { s 2 , s 3 , s 4 } \{s_2,s_3,s_4\} { s 2 , s 3 , s 4 } { s 1 , s 2 , s 4 } \{s_1,s_2,s_4\} { s 1 , s 2 , s 4 } { s 1 , s 3 , s 4 } \{s_1,s_3,s_4\} { s 1 , s 3 , s 4 } { u , φ u , φ 2 u } \{u,\varphi u,\varphi^2u\} { u , φ u , φ 2 u } { u , φ u , φ 2 u } \{u,\varphi u,\varphi^2u\} { u , φ u , φ 2 u } { φ u , φ 2 u , φ 3 u } \{\varphi u,\varphi^2u,\varphi^3u\} { φ u , φ 2 u , φ 3 u } { φ u , φ 2 u , φ 3 u } \{\varphi u,\varphi^2u,\varphi^3u\} { φ u , φ 2 u , φ 3 u }
(c) S S S r S ( φ u ) = 2 r_S(\varphi u)=2 r S ( φ u ) = 2 ∣ S − S ∣ = 2 ∣ Δ ( S ) ∣ + 1 = 9 |S-S|=2|\Delta(S)|+1=9 ∣ S − S ∣ = 2∣Δ ( S ) ∣ + 1 = 9 A ⊂ R A\subset\mathbb{R} A ⊂ R 7 ≤ ∣ A − A ∣ ≤ 13 7\leq|A-A|\leq13 7 ≤ ∣ A − A ∣ ≤ 13 [41 , 52 ] . So S S S
(d) The difference multiset does not depend on F F F F > a + b F>a+b F > a + b
Maximality and uniqueness
Theorem 2. Let u > 0 u>0 u > 0 D = { u , φ u , φ 2 u , φ 3 u } D=\{u,\varphi u,\varphi^{2}u,\varphi^{3}u\} D = { u , φ u , φ 2 u , φ 3 u }
(i) There do not exist five (or more) distinct real numbers all of whose positive differences belong to D D D
(ii) If A ⊂ R A\subset\mathbb{R} A ⊂ R Δ ( A ) ⊆ D \Delta(A)\subseteq D Δ ( A ) ⊆ D A = { y , y + φ u , y + φ 2 u , y + φ 3 u } A=\{y,y+\varphi u,y+\varphi^{2}u,y+\varphi^{3}u\} A = { y , y + φ u , y + φ 2 u , y + φ 3 u } y = min A y=\min A y = min A A A A S ( F , u ) S(F,u) S ( F , u ) F F F Δ ( A ) = D \Delta(A)=D Δ ( A ) = D r A = r S r_A=r_S r A = r S
Hence max { ∣ A ∣ : A ⊆ R , Δ ( A ) ⊆ D } = 4 \max\{|A|:A\subseteq\mathbb{R},\Delta(A)\subseteq D\}=4 max { ∣ A ∣ : A ⊆ R , Δ ( A ) ⊆ D } = 4 S S S
Proof. Let y 1 < ⋯ < y m y_1<\cdots<y_m y 1 < ⋯ < y m D D D g i = y i + 1 − y i g_i=y_{i+1}-y_i g i = y i + 1 − y i 1 ≤ i < m 1\leq i<m 1 ≤ i < m g i ∈ D g_i\in D g i ∈ D g i ≥ u g_i\geq u g i ≥ u y m − y 1 = g 1 + ⋯ + g m − 1 y_m-y_1=g_1+\cdots+g_{m-1} y m − y 1 = g 1 + ⋯ + g m − 1 D D D
(i) Suppose m ≥ 5 m\geq5 m ≥ 5 y 1 < ⋯ < y 5 y_1<\cdots<y_5 y 1 < ⋯ < y 5 m = 5 m=5 m = 5 y 5 − y 1 ≥ 4 u y_5-y_1\geq4u y 5 − y 1 ≥ 4 u Lemma 2.4 (iii), φ 2 u < 3 u < 4 u < φ 3 u \varphi^{2}u<3u<4u<\varphi^{3}u φ 2 u < 3 u < 4 u < φ 3 u D D D ≥ 4 u \geq4u ≥ 4 u φ 3 u \varphi^{3}u φ 3 u g 1 + g 2 + g 3 + g 4 = φ 3 u = ( 2 φ + 1 ) u g_1+g_2+g_3+g_4=\varphi^{3}u=(2\varphi+1)u g 1 + g 2 + g 3 + g 4 = φ 3 u = ( 2 φ + 1 ) u g i = u g_i=u g i = u 4 = 2 φ + 1 4=2\varphi+1 4 = 2 φ + 1 φ = 3 / 2 \varphi=3/2 φ = 3/2 Lemma 2.4 (iii). Otherwise some g i ≥ φ u g_i\geq\varphi u g i ≥ φ u ( 3 + φ ) u > ( 2 φ + 1 ) u (3+\varphi)u>(2\varphi+1)u ( 3 + φ ) u > ( 2 φ + 1 ) u φ < 2 \varphi<2 φ < 2
(ii) Now m = 4 m=4 m = 4 ≥ 3 u > φ 2 u \geq3u>\varphi^{2}u ≥ 3 u > φ 2 u g 1 + g 2 + g 3 = φ 3 u g_1+g_2+g_3=\varphi^{3}u g 1 + g 2 + g 3 = φ 3 u φ 3 u \varphi^{3}u φ 3 u φ 2 u \varphi^{2}u φ 2 u ( φ 3 − φ 2 ) u = φ u < 2 u (\varphi^{3}-\varphi^{2})u=\varphi u<2u ( φ 3 − φ 2 ) u = φ u < 2 u ≥ u \geq u ≥ u { u , φ u } \{u,\varphi u\} { u , φ u } k k k φ u \varphi u φ u ( 3 − k ) + k φ = 1 + 2 φ (3-k)+k\varphi=1+2\varphi ( 3 − k ) + k φ = 1 + 2 φ ( 2 − k ) ( 1 − φ ) = 0 (2-k)(1-\varphi)=0 ( 2 − k ) ( 1 − φ ) = 0 k = 2 k=2 k = 2 ( g 1 , g 2 , g 3 ) (g_1,g_2,g_3) ( g 1 , g 2 , g 3 ) ( u , φ u , φ u ) (u,\varphi u,\varphi u) ( u , φ u , φ u ) ( g 1 , g 2 , g 3 ) = ( u , φ u , φ u ) (g_1,g_2,g_3)=(u,\varphi u,\varphi u) ( g 1 , g 2 , g 3 ) = ( u , φ u , φ u ) y 4 − y 2 = 2 φ u y_4-y_2=2\varphi u y 4 − y 2 = 2 φ u ( g 1 , g 2 , g 3 ) = ( φ u , φ u , u ) (g_1,g_2,g_3)=(\varphi u,\varphi u,u) ( g 1 , g 2 , g 3 ) = ( φ u , φ u , u ) y 3 − y 1 = 2 φ u y_3-y_1=2\varphi u y 3 − y 1 = 2 φ u 2 φ u ∉ D 2\varphi u\notin D 2 φ u ∈ / D 2 φ ≠ 1 2\varphi\neq1 2 φ = 1 2 φ ≠ φ 2\varphi\neq\varphi 2 φ = φ 2 φ ≠ φ + 1 2\varphi\neq\varphi+1 2 φ = φ + 1 φ ≠ 1 \varphi\neq1 φ = 1 2 φ ≠ 2 φ + 1 2\varphi\neq2\varphi+1 2 φ = 2 φ + 1 ( g 1 , g 2 , g 3 ) = ( φ u , u , φ u ) (g_1,g_2,g_3)=(\varphi u,u,\varphi u) ( g 1 , g 2 , g 3 ) = ( φ u , u , φ u ) A = { y 1 , y 1 + φ u , y 1 + ( φ + 1 ) u , y 1 + ( 2 φ + 1 ) u } = { y 1 , y 1 + φ u , y 1 + φ 2 u , y 1 + φ 3 u } A=\{y_1,y_1+\varphi u,y_1+(\varphi+1)u,y_1+(2\varphi+1)u\}=\{y_1,y_1+\varphi u,y_1+\varphi^{2}u,y_1+\varphi^{3}u\} A = { y 1 , y 1 + φ u , y 1 + ( φ + 1 ) u , y 1 + ( 2 φ + 1 ) u } = { y 1 , y 1 + φ u , y 1 + φ 2 u , y 1 + φ 3 u } A = S ( F , u ) + ( y 1 − F + φ 3 u ) A=S(F,u)+(y_1-F+\varphi^{3}u) A = S ( F , u ) + ( y 1 − F + φ 3 u ) Δ ( A ) = D \Delta(A)=D Δ ( A ) = D r A = r S r_A=r_S r A = r S
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Remark 4.1. A counting argument alone gives only ∣ A ∣ ≤ 5 |A|\leq5 ∣ A ∣ ≤ 5 y i − y 1 y_i-y_1 y i − y 1 2 ≤ i ≤ m 2\leq i\leq m 2 ≤ i ≤ m m − 1 m-1 m − 1 D D D
The same conclusion holds for the whole multiplicative orbit { c φ n : n ∈ Z } \{c\varphi^n:n\in\mathbb{Z}\} { c φ n : n ∈ Z }
Lemma 4.2. Let p , q , r ∈ Z p,q,r\in\mathbb{Z} p , q , r ∈ Z φ p + φ q = φ r \varphi^p+\varphi^q=\varphi^r φ p + φ q = φ r { p , q } = { r − 1 , r − 2 } \{p,q\}=\{r-1,r-2\} { p , q } = { r − 1 , r − 2 } p ≠ q p\neq q p = q ∣ p − q ∣ = 1 |p-q|=1 ∣ p − q ∣ = 1
Proof. “If” is Lemma 2.4 (ii). Conversely, assume p ≥ q p\geq q p ≥ q α = p − r ≥ β = q − r \alpha=p-r\geq\beta=q-r α = p − r ≥ β = q − r φ α + φ β = 1 \varphi^\alpha+\varphi^\beta=1 φ α + φ β = 1 α ≥ 0 \alpha\geq0 α ≥ 0 α ≤ − 2 \alpha\leq-2 α ≤ − 2 φ α + φ β ≤ 2 φ − 2 = 2 ( 2 − φ ) = 4 − 2 φ < 1 \varphi^\alpha+\varphi^\beta\leq2\varphi^{-2}=2(2-\varphi)=4-2\varphi<1 φ α + φ β ≤ 2 φ − 2 = 2 ( 2 − φ ) = 4 − 2 φ < 1 φ > 3 / 2 \varphi>3/2 φ > 3/2 α = − 1 \alpha=-1 α = − 1 φ β = 1 − φ − 1 = 2 − φ = φ − 2 \varphi^\beta=1-\varphi^{-1}=2-\varphi=\varphi^{-2} φ β = 1 − φ − 1 = 2 − φ = φ − 2 Lemma 2.4 (ii)), so β = − 2 \beta=-2 β = − 2 Lemma 2.4 (vi).
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Remark 4.3. Lemma 4.2 is classical. Nagell [37 Théorème 1] states that in a quadratic field a relation 1 + ξ + ξ 1 = 0 1+\xi+\xi_1=0 1 + ξ + ξ 1 = 0 Q ( − 3 ) \mathbb{Q}(\sqrt{-3}) Q ( − 3 ) Q ( 5 ) \mathbb{Q}(\sqrt{5}) Q ( 5 ) Q ( 5 ) \mathbb{Q}(\sqrt{5}) Q ( 5 ) 1 + ξ − ξ 2 = 0 1+\xi-\xi^2=0 1 + ξ − ξ 2 = 0 1 − ξ + ξ − 1 = 0 1-\xi+\xi^{-1}=0 1 − ξ + ξ − 1 = 0 1 − ξ − 1 − ξ − 2 = 0 1-\xi^{-1}-\xi^{-2}=0 1 − ξ − 1 − ξ − 2 = 0 ξ = φ \xi=\varphi ξ = φ φ ˉ \bar{\varphi} φ ˉ φ p − r + φ q − r = 1 \varphi^{p-r}+\varphi^{q-r}=1 φ p − r + φ q − r = 1 ξ + ξ ′ = 1 \xi+\xi'=1 ξ + ξ ′ = 1 Z [ φ ] × \mathbb{Z}[\varphi]^\times Z [ φ ] × ξ \xi ξ 1 − ξ 1-\xi 1 − ξ exceptional ; Nagell determined the exceptional units of all number fields with unit rank at most one (see [39 §1, no. 2] and the works cited there, and [30 §3] ). In Q ( 5 ) \mathbb{Q}(\sqrt{5}) Q ( 5 ) φ \varphi φ φ 2 \varphi^2 φ 2 φ − 1 \varphi^{-1} φ − 1 φ − 2 \varphi^{-2} φ − 2 − φ -\varphi − φ − φ − 1 -\varphi^{-1} − φ − 1 1 − φ = − φ − 1 1-\varphi=-\varphi^{-1} 1 − φ = − φ − 1 1 − φ 2 = − φ 1-\varphi^2=-\varphi 1 − φ 2 = − φ 1 − φ − 1 = φ − 2 1-\varphi^{-1}=\varphi^{-2} 1 − φ − 1 = φ − 2 1 − φ − 2 = φ − 1 1-\varphi^{-2}=\varphi^{-1} 1 − φ − 2 = φ − 1 1 + φ = φ 2 1+\varphi=\varphi^2 1 + φ = φ 2 1 + φ − 1 = φ 1+\varphi^{-1}=\varphi 1 + φ − 1 = φ ξ = ± φ p \xi=\pm\varphi^p ξ = ± φ p 1 − ξ = ± φ q 1-\xi=\pm\varphi^q 1 − ξ = ± φ q ξ + ( 1 − ξ ) = 1 \xi+(1-\xi)=1 ξ + ( 1 − ξ ) = 1 φ p + φ q = φ 0 \varphi^p+\varphi^q=\varphi^0 φ p + φ q = φ 0 φ 0 + φ q = φ p \varphi^0+\varphi^q=\varphi^p φ 0 + φ q = φ p φ 0 + φ p = φ q \varphi^0+\varphi^p=\varphi^q φ 0 + φ p = φ q − φ p − φ q = 1 -\varphi^p-\varphi^q=1 − φ p − φ q = 1 Theorem 2 ′(ii): after translating and dilating so that two of the points are 0 and 1 (Lenstra’s normalisation [30 §2, p. 243] ; cf. [31 ] ), each remaining point ξ \xi ξ
Theorem 2′. Let c > 0 c>0 c > 0 Φ c = { c φ n : n ∈ Z } \Phi_c=\{c\varphi^n:n\in\mathbb{Z}\} Φ c = { c φ n : n ∈ Z }
(i) A three-element set { y 1 < y 2 < y 3 } \{y_1<y_2<y_3\} { y 1 < y 2 < y 3 } Δ ( { y 1 , y 2 , y 3 } ) ⊆ Φ c \Delta(\{y_1,y_2,y_3\})\subseteq\Phi_c Δ ({ y 1 , y 2 , y 3 }) ⊆ Φ c ( c φ n + 1 , c φ n ) (c\varphi^{n+1},c\varphi^n) ( c φ n + 1 , c φ n ) ( c φ n , c φ n + 1 ) (c\varphi^n,c\varphi^{n+1}) ( c φ n , c φ n + 1 ) n ∈ Z n\in\mathbb{Z} n ∈ Z
(ii) A four-element set A A A Δ ( A ) ⊆ Φ c \Delta(A)\subseteq\Phi_c Δ ( A ) ⊆ Φ c A = t + c φ n { 0 , φ , φ 2 , φ 3 } A=t+c\varphi^n\{0,\varphi,\varphi^2,\varphi^3\} A = t + c φ n { 0 , φ , φ 2 , φ 3 } t ∈ R t\in\mathbb{R} t ∈ R n ∈ Z n\in\mathbb{Z} n ∈ Z A A A S ( F , c φ n ) S(F,c\varphi^n) S ( F , c φ n )
(iii) No set of five or more distinct reals has all its positive differences in Φ c \Phi_c Φ c
In particular Theorem 2 follows again: if Δ ( A ) ⊆ D ( u ) ⊆ Φ u \Delta(A)\subseteq D(u)\subseteq\Phi_u Δ ( A ) ⊆ D ( u ) ⊆ Φ u ∣ A ∣ = 4 |A|=4 ∣ A ∣ = 4 A = t + u φ n { 0 , φ , φ 2 , φ 3 } A=t+u\varphi^n\{0,\varphi,\varphi^2,\varphi^3\} A = t + u φ n { 0 , φ , φ 2 , φ 3 } Δ ( A ) = { u φ n , … , u φ n + 3 } ⊆ D ( u ) \Delta(A)=\{u\varphi^n,\ldots,u\varphi^{n+3}\}\subseteq D(u) Δ ( A ) = { u φ n , … , u φ n + 3 } ⊆ D ( u ) n = 0 n=0 n = 0
Proof. Write the gaps of an increasing set as g i = c φ e i g_i=c\varphi^{e_i} g i = c φ e i
(i) The differences are g 1 g_1 g 1 g 2 g_2 g 2 g 1 + g 2 g_1+g_2 g 1 + g 2 φ e 1 + φ e 2 = φ r \varphi^{e_1}+\varphi^{e_2}=\varphi^r φ e 1 + φ e 2 = φ r r r r { e 1 , e 2 } = { r − 1 , r − 2 } \{e_1,e_2\}=\{r-1,r-2\} { e 1 , e 2 } = { r − 1 , r − 2 } n = r − 2 n=r-2 n = r − 2 g 1 + g 2 = c φ n + 2 g_1+g_2=c\varphi^{n+2} g 1 + g 2 = c φ n + 2
(ii) Let A = { y 1 < y 2 < y 3 < y 4 } A=\{y_1<y_2<y_3<y_4\} A = { y 1 < y 2 < y 3 < y 4 } Δ ( A ) ⊆ Φ c \Delta(A)\subseteq\Phi_c Δ ( A ) ⊆ Φ c g 1 + g 2 = c φ r g_1+g_2=c\varphi^r g 1 + g 2 = c φ r { y 1 , y 2 , y 3 } \{y_1,y_2,y_3\} { y 1 , y 2 , y 3 } { y 1 , y 3 , y 4 } \{y_1,y_3,y_4\} { y 1 , y 3 , y 4 } { y 2 , y 3 , y 4 } \{y_2,y_3,y_4\} { y 2 , y 3 , y 4 } { e 1 , e 2 } = { r − 1 , r − 2 } \{e_1,e_2\}=\{r-1,r-2\} { e 1 , e 2 } = { r − 1 , r − 2 } e 3 ∈ { r − 1 , r + 1 } e_3\in\{r-1,r+1\} e 3 ∈ { r − 1 , r + 1 } ∣ e 2 − e 3 ∣ = 1 |e_2-e_3|=1 ∣ e 2 − e 3 ∣ = 1 e 3 = r + 1 e_3=r+1 e 3 = r + 1 ∣ e 2 − e 3 ∣ ≥ 2 |e_2-e_3|\geq2 ∣ e 2 − e 3 ∣ ≥ 2 e 3 = r − 1 e_3=r-1 e 3 = r − 1 ∣ e 2 − ( r − 1 ) ∣ = 1 |e_2-(r-1)|=1 ∣ e 2 − ( r − 1 ) ∣ = 1 e 2 ∈ { r − 1 , r − 2 } e_2\in\{r-1,r-2\} e 2 ∈ { r − 1 , r − 2 } e 2 = r − 2 e_2=r-2 e 2 = r − 2 e 1 = r − 1 e_1=r-1 e 1 = r − 1 n = r − 2 n=r-2 n = r − 2 ( c φ n + 1 , c φ n , c φ n + 1 ) (c\varphi^{n+1},c\varphi^n,c\varphi^{n+1}) ( c φ n + 1 , c φ n , c φ n + 1 ) A = y 1 + c φ n { 0 , φ , φ + 1 , 2 φ + 1 } = y 1 + c φ n { 0 , φ , φ 2 , φ 3 } A=y_1+c\varphi^n\{0,\varphi,\varphi+1,2\varphi+1\}=y_1+c\varphi^n\{0,\varphi,\varphi^2,\varphi^3\} A = y 1 + c φ n { 0 , φ , φ + 1 , 2 φ + 1 } = y 1 + c φ n { 0 , φ , φ 2 , φ 3 } Δ ( A ) = { c φ n , … , c φ n + 3 } ⊆ Φ c \Delta(A)=\{c\varphi^n,\ldots,c\varphi^{n+3}\}\subseteq\Phi_c Δ ( A ) = { c φ n , … , c φ n + 3 } ⊆ Φ c u = c φ n u=c\varphi^n u = c φ n S ( F , c φ n ) S(F,c\varphi^n) S ( F , c φ n )
(iii) Let y 1 < ⋯ < y 5 y_1<\cdots<y_5 y 1 < ⋯ < y 5 Φ c \Phi_c Φ c { y 1 , … , y 4 } \{y_1,\ldots,y_4\} { y 1 , … , y 4 } s s s ( e 1 , e 2 , e 3 ) = ( s + 1 , s , s + 1 ) (e_1,e_2,e_3)=(s+1,s,s+1) ( e 1 , e 2 , e 3 ) = ( s + 1 , s , s + 1 ) { y 2 , … , y 5 } \{y_2,\ldots,y_5\} { y 2 , … , y 5 } s ′ s' s ′ ( e 2 , e 3 , e 4 ) = ( s ′ + 1 , s ′ , s ′ + 1 ) (e_2,e_3,e_4)=(s'+1,s',s'+1) ( e 2 , e 3 , e 4 ) = ( s ′ + 1 , s ′ , s ′ + 1 ) e 2 = s = s ′ + 1 e_2=s=s'+1 e 2 = s = s ′ + 1 e 3 = s + 1 = s ′ e_3=s+1=s' e 3 = s + 1 = s ′ s = s ′ + 1 = s ′ + 2 s=s'+1=s'+2 s = s ′ + 1 = s ′ + 2
Part (i) of Theorem 2 ′ is a restatement of Lemma 4.2 and hence classical (Remark 4.3). Part (iii) is known, see Remark 4.5.
Definition 4.4. For x > 1 x>1 x > 1 N ( x ) ∈ { 2 , 3 , … } ∪ { ∞ } N(x)\in\{2,3,\ldots\}\cup\{\infty\} N ( x ) ∈ { 2 , 3 , … } ∪ { ∞ } A ⊂ R A\subset\mathbb{R} A ⊂ R { c x n : n ∈ Z } \{cx^n:n\in\mathbb{Z}\} { c x n : n ∈ Z } c > 0 c>0 c > 0
Remark 4.5 (relation to Lenstra’s constant). The bound in Theorem 2 ′(iii), and hence in Theorem 2 (i), i.e. N ( φ ) ≤ 4 N(\varphi) \le4 N ( φ ) ≤ 4 N ( ϱ ) ≤ 5 N(\varrho) \le5 N ( ϱ ) ≤ 5 ϱ \varrho ϱ ϱ 3 = ϱ + 1 \varrho^{3} = \varrho+ 1 ϱ 3 = ϱ + 1 § 8 ). We recall the argument, which is that of Lenstra [30 (2.2)] . Let A A A { c x n : n ∈ Z } \{cx^{n}: n \in\mathbb{Z}\} { c x n : n ∈ Z } x ∈ { φ , ϱ } x \in\{\varphi,\varrho\} x ∈ { φ , ϱ } A ′ = c − 1 ( A − min A ) A' = c^{-1}(A-\min A) A ′ = c − 1 ( A − min A ) A ′ A' A ′ x e x^{e} x e e ∈ Z e \in\mathbb{Z} e ∈ Z φ − 1 = φ − 1 \varphi^{-1} = \varphi- 1 φ − 1 = φ − 1 ϱ − 1 = ϱ 2 − 1 \varrho^{-1} = \varrho^{2} - 1 ϱ − 1 = ϱ 2 − 1 x x x Z [ x ] \mathbb{Z}[x] Z [ x ] A ′ ⊂ Z [ x ] A' \subset\mathbb{Z}[x] A ′ ⊂ Z [ x ] A ′ A' A ′ ± x n \pm x^{n} ± x n Z [ x ] \mathbb{Z}[x] Z [ x ] ψ \psi ψ Z [ x ] \mathbb{Z}[x] Z [ x ] ψ \psi ψ A ′ A' A ′ ∣ A ′ ∣ |A'| ∣ A ′ ∣ X 2 − X − 1 ≡ X 2 + X + 1 ( m o d 2 ) X^{2} - X - 1 \equiv X^{2} + X + 1 \pmod{2} X 2 − X − 1 ≡ X 2 + X + 1 ( mod 2 ) φ ↦ ω \varphi\mapsto\omega φ ↦ ω Z [ φ ] ≅ Z [ X ] / ( X 2 − X − 1 ) → F 4 = F 2 [ ω ] \mathbb{Z}[\varphi] \cong\mathbb{Z}[X]/(X^{2} - X - 1) \to\mathbb{F}_{4} = \mathbb{F}_{2}[\omega] Z [ φ ] ≅ Z [ X ] / ( X 2 − X − 1 ) → F 4 = F 2 [ ω ] ω 2 + ω + 1 = 0 \omega^{2} + \omega+ 1 = 0 ω 2 + ω + 1 = 0 N ( φ ) ≤ 4 N(\varphi) \le4 N ( φ ) ≤ 4 2 3 − 2 − 1 = 5 2^{3} - 2 - 1 = 5 2 3 − 2 − 1 = 5 ϱ ↦ 2 \varrho\mapsto2 ϱ ↦ 2 Z [ ϱ ] ≅ Z [ X ] / ( X 3 − X − 1 ) → F 5 \mathbb{Z}[\varrho] \cong\mathbb{Z}[X]/(X^{3} - X - 1) \to\mathbb{F}_{5} Z [ ϱ ] ≅ Z [ X ] / ( X 3 − X − 1 ) → F 5 X 3 − X − 1 ≡ ( X − 2 ) ( X 2 + 2 X + 3 ) ( m o d 5 ) X^{3} - X - 1 \equiv(X - 2)(X^{2} + 2X + 3) \pmod{5} X 3 − X − 1 ≡ ( X − 2 ) ( X 2 + 2 X + 3 ) ( mod 5 ) − 8 ≡ 2 -8 \equiv2 − 8 ≡ 2 N ( ϱ ) ≤ 5 N(\varrho) \le5 N ( ϱ ) ≤ 5 [30 (2.1)] , L = 4 L = 4 L = 4 Q ( 5 ) \mathbb{Q}(\sqrt{5}) Q ( 5 ) L = 5 L = 5 L = 5 Q ( ϱ ) \mathbb{Q}(\varrho) Q ( ϱ ) M ≤ L M \le L M ≤ L [30 (2.2)] .) Together with Theorem 1 and Proposition 8.1 this gives N ( φ ) = 4 N(\varphi) = 4 N ( φ ) = 4 N ( ϱ ) = 5 N(\varrho) = 5 N ( ϱ ) = 5 M = 4 M = 4 M = 4 Q ( 5 ) \mathbb{Q}(\sqrt{5}) Q ( 5 ) [30 (3.9)] , obtained there by applying [30 (2.4)(b)] with f = X 2 − X − 1 f = X^{2} - X - 1 f = X 2 − X − 1 0 , 1 , φ , φ + 1 = φ 2 0,1,\varphi,\varphi+ 1 = \varphi^{2} 0 , 1 , φ , φ + 1 = φ 2 φ − 1 { 0 , φ , φ 2 , φ 3 } \varphi^{-1}\{0,\varphi,\varphi^{2},\varphi^{3}\} φ − 1 { 0 , φ , φ 2 , φ 3 } M = 5 M = 5 M = 5 K = Q ( α ) K = \mathbb{Q}(\alpha) K = Q ( α ) α 3 − α − 1 = 0 \alpha^{3} - \alpha- 1 = 0 α 3 − α − 1 = 0 Δ = − 23 \Delta= -23 Δ = − 23 [30 (3.10)] , i.e. K = Q ( ϱ ) K = \mathbb{Q}(\varrho) K = Q ( ϱ ) disc ( X 3 − X − 1 ) = − 23 \operatorname{disc}(X^{3} - X - 1) = -23 disc ( X 3 − X − 1 ) = − 23 Z [ ϱ ] \mathbb{Z}[\varrho] Z [ ϱ ] [30 ] is the uniqueness statement of Theorem 2 (ii), the explicit uniqueness statement of Theorem 2 ′(ii) (see above), the rigidity Theorem 3 ′, and the analysis of other bases x x x § 5 ).
Uniqueness of the base and closure residues
Definition 5.1. Let x > 1 x > 1 x > 1 u > 0 u > 0 u > 0 F > ( x + x 2 ) u F > (x + x^{2})u F > ( x + x 2 ) u x x x
S x = ( F , F − x u , F − x 2 u , F − ( x + x 2 ) u ) , S_x = (F, F - xu, F - x^{2}u, F - (x + x^{2})u), S x = ( F , F − xu , F − x 2 u , F − ( x + x 2 ) u ) , and its target is D x = { u , x u , x 2 u , x 3 u } D_x = \{u, xu, x^{2}u, x^{3}u\} D x = { u , xu , x 2 u , x 3 u } P ( x ) = x 2 − x − 1 P(x) = x^{2} - x - 1 P ( x ) = x 2 − x − 1
ρ − ( x ) = ( x 2 u − x u ) − u = P ( x ) u , ρ + ( x ) = x 3 u − ( x u + x 2 u ) = x P ( x ) u . \rho_{-}(x) = (x^{2}u - xu) - u = P(x)u,\qquad\rho_{+}(x) = x^{3}u - (xu + x^{2}u) = xP(x)u. ρ − ( x ) = ( x 2 u − xu ) − u = P ( x ) u , ρ + ( x ) = x 3 u − ( xu + x 2 u ) = x P ( x ) u . The residue ρ − \rho_{-} ρ − b − a = u b-a = u b − a = u ρ + \rho_{+} ρ + a + b = x 3 u a+b = x^{3}u a + b = x 3 u a = x u a = xu a = xu b = x 2 u b = x^{2}u b = x 2 u Lemma 2.4 (v), S φ = S S_{\varphi} = S S φ = S
Theorem 3. Let x > 1 x > 1 x > 1
(a) The difference multiset of S x S_x S x
{ ( x 2 − x ) u , x u , x u , x 2 u , x 2 u , ( x 2 + x ) u } = { u + ρ − ( x ) , x u , x u , x 2 u , x 2 u , x 3 u − ρ + ( x ) } . \{(x^{2} - x)u, xu, xu, x^{2}u, x^{2}u, (x^{2} + x)u\} = \{u + \rho_{-}(x), xu, xu, x^{2}u, x^{2}u, x^{3}u - \rho_{+}(x)\}. {( x 2 − x ) u , xu , xu , x 2 u , x 2 u , ( x 2 + x ) u } = { u + ρ − ( x ) , xu , xu , x 2 u , x 2 u , x 3 u − ρ + ( x )} . (b) Δ ( S x ) = D x ⟺ ρ − ( x ) = 0 ⟺ ρ + ( x ) = 0 ⟺ x = φ \Delta(S_x) = D_x \Longleftrightarrow\rho_{-}(x) = 0 \Longleftrightarrow\rho_{+}(x) = 0 \Longleftrightarrow x = \varphi Δ ( S x ) = D x ⟺ ρ − ( x ) = 0 ⟺ ρ + ( x ) = 0 ⟺ x = φ
(c) If x ≠ φ x \ne\varphi x = φ ρ − ( x ) ≠ 0 \rho_{-}(x) \ne0 ρ − ( x ) = 0 ρ + ( x ) ≠ 0 \rho_{+}(x) \ne0 ρ + ( x ) = 0 x − φ x - \varphi x − φ ρ + ( x ) = x ρ − ( x ) \rho_{+}(x) = x\rho_{-}(x) ρ + ( x ) = x ρ − ( x )
∣ ρ − ( x ) ∣ = ∣ x − φ ∣ ( x − φ ˉ ) u . |\rho_{-}(x)| = |x - \varphi|(x - \bar{\varphi})u. ∣ ρ − ( x ) ∣ = ∣ x − φ ∣ ( x − φ ˉ ) u . (d) More strongly, Δ ( S x ) \Delta(S_x) Δ ( S x ) x = φ x = \varphi x = φ x ≠ φ x \ne\varphi x = φ c > 0 c > 0 c > 0 k ∈ Z k \in\mathbb{Z} k ∈ Z Δ ( S x ) = { c x k , c x k + 1 , c x k + 2 , c x k + 3 } \Delta(S_x) = \{cx^{k}, cx^{k+1}, cx^{k+2}, cx^{k+3}\} Δ ( S x ) = { c x k , c x k + 1 , c x k + 2 , c x k + 3 }
Proof. (a) Apply Lemma 3.1 with α = x u < β = x 2 u \alpha=xu<\beta=x^{2}u α = xu < β = x 2 u S x = F − Q S_x=F-Q S x = F − Q β − α = ( x 2 − x ) u \beta-\alpha=(x^{2}-x)u β − α = ( x 2 − x ) u α + β = ( x 2 + x ) u \alpha+\beta=(x^{2}+x)u α + β = ( x 2 + x ) u ρ ± \rho_{\pm} ρ ±
(b) The last three conditions are equivalent because ρ − = P ( x ) u \rho_{-}=P(x)u ρ − = P ( x ) u ρ + = x P ( x ) u \rho_{+}=xP(x)u ρ + = x P ( x ) u x , u > 0 x,u>0 x , u > 0 P ( x ) = 0 ⟺ x = φ P(x)=0\Longleftrightarrow x=\varphi P ( x ) = 0 ⟺ x = φ x > 1 x>1 x > 1 Lemma 2.4 (v)). If x = φ x=\varphi x = φ Δ ( S φ ) = D φ \Delta(S_{\varphi})=D_{\varphi} Δ ( S φ ) = D φ Δ ( S x ) = D x \Delta(S_x)=D_x Δ ( S x ) = D x x = 2 x=2 x = 2 ( x 2 − x ) u = x u (x^{2}-x)u=xu ( x 2 − x ) u = xu ∣ Δ ( S x ) ∣ = 3 < 4 = ∣ D x ∣ |\Delta(S_x)|=3<4=|D_x| ∣Δ ( S x ) ∣ = 3 < 4 = ∣ D x ∣ x ≠ 2 x\ne2 x = 2 ( x 2 − x ) u (x^{2}-x)u ( x 2 − x ) u x u xu xu x 2 u x^{2}u x 2 u ( x 2 + x ) u (x^{2}+x)u ( x 2 + x ) u x 2 − x ≠ x x^{2}-x\ne x x 2 − x = x x ≠ 2 x\ne2 x = 2 x 2 − x < x 2 < x 2 + x x^{2}-x<x^{2}<x^{2}+x x 2 − x < x 2 < x 2 + x x < x 2 x<x^{2} x < x 2 ( x 2 ∓ x ) u (x^{2}\mp x)u ( x 2 ∓ x ) u { x u , x 2 u } \{xu,x^{2}u\} { xu , x 2 u } u , x 3 u u,x^{3}u u , x 3 u { x u , x 2 u } \{xu,x^{2}u\} { xu , x 2 u } x u , x 2 u xu,x^{2}u xu , x 2 u { ( x 2 − x ) u , ( x 2 + x ) u } = { u , x 3 u } \{(x^{2}-x)u,(x^{2}+x)u\}=\{u,x^{3}u\} {( x 2 − x ) u , ( x 2 + x ) u } = { u , x 3 u } x 2 − x = 1 x^{2}-x=1 x 2 − x = 1 x 2 + x = x 3 x^{2}+x=x^{3} x 2 + x = x 3 P ( x ) = 0 P(x)=0 P ( x ) = 0 x = φ x=\varphi x = φ
(c) This is Lemma 2.4 (v) together with ρ − = P ( x ) u \rho_{-}=P(x)u ρ − = P ( x ) u ρ + = x ρ − \rho_{+}=x\rho_{-} ρ + = x ρ −
(d) “If” is Theorem 1. Suppose Δ ( S x ) \Delta(S_x) Δ ( S x ) ∣ Δ ( S x ) ∣ = 4 |\Delta(S_x)|=4 ∣Δ ( S x ) ∣ = 4 x ≠ 2 x\ne2 x = 2
Case 1 < x < 2 1<x<2 1 < x < 2 Here x 2 − x < x x^{2}-x<x x 2 − x < x ( x 2 − x ) u < x u < x 2 u < ( x 2 + x ) u (x^{2}-x)u<xu<x^{2}u<(x^{2}+x)u ( x 2 − x ) u < xu < x 2 u < ( x 2 + x ) u 1 x − 1 \frac{1}{x-1} x − 1 1 x x x x + 1 x \frac{x+1}{x} x x + 1 x ( x − 1 ) = 1 x(x-1)=1 x ( x − 1 ) = 1 P ( x ) = 0 P(x)=0 P ( x ) = 0 x = φ x=\varphi x = φ x + 1 x = x \frac{x+1}{x}=x x x + 1 = x
Case x > 2 x>2 x > 2 Here x < x 2 − x x<x^{2}-x x < x 2 − x x u < ( x 2 − x ) u < x 2 u < ( x 2 + x ) u xu<(x^{2}-x)u<x^{2}u<(x^{2}+x)u xu < ( x 2 − x ) u < x 2 u < ( x 2 + x ) u x − 1 x-1 x − 1 x x − 1 \frac{x}{x-1} x − 1 x x + 1 x \frac{x+1}{x} x x + 1 ( x − 1 ) 2 = x (x-1)^{2}=x ( x − 1 ) 2 = x x 2 − 3 x + 1 = 0 x^{2}-3x+1=0 x 2 − 3 x + 1 = 0 ( 3 + 5 ) / 2 = φ 2 (3+\sqrt{5})/2=\varphi^{2} ( 3 + 5 ) /2 = φ 2 x ( x − 1 ) = x + 1 x(x-1)=x+1 x ( x − 1 ) = x + 1 x 2 − 2 x − 1 = 0 x^{2}-2x-1=0 x 2 − 2 x − 1 = 0 1 + 2 1+\sqrt{2} 1 + 2 φ 2 = 1 + 2 \varphi^{2}=1+\sqrt{2} φ 2 = 1 + 2 1 + 5 = 2 2 1+\sqrt{5}=2\sqrt{2} 1 + 5 = 2 2 6 + 2 5 = 8 6+2\sqrt{5}=8 6 + 2 5 = 8 5 = 1 \sqrt{5}=1 5 = 1 x > 2 x>2 x > 2
The final assertion holds because { c x k , … , c x k + 3 } \{cx^{k},\ldots,cx^{k+3}\} { c x k , … , c x k + 3 } x x x
Fibonacci quotients
Lemma 5.2 (Cassini’s identity). F i b n + 1 F i b n − 1 − F i b n 2 = ( − 1 ) n \mathrm{Fib}_{n+1}\mathrm{Fib}_{n-1}-\mathrm{Fib}_{n}^{2}=(-1)^{n} Fib n + 1 Fib n − 1 − Fib n 2 = ( − 1 ) n n ≥ 1 n\ge1 n ≥ 1
Proof. For n = 1 n=1 n = 1 F i b 2 F i b 0 − F i b 1 2 = − 1 \mathrm{Fib}_{2}\mathrm{Fib}_{0}-\mathrm{Fib}_{1}^{2}=-1 Fib 2 Fib 0 − Fib 1 2 = − 1 c n = F i b n + 1 F i b n − 1 − F i b n 2 c_n=\mathrm{Fib}_{n+1}\mathrm{Fib}_{n-1}-\mathrm{Fib}_{n}^{2} c n = Fib n + 1 Fib n − 1 − Fib n 2 c n + 1 = F i b n + 2 F i b n − F i b n + 1 2 = ( F i b n + 1 + F i b n ) F i b n − F i b n + 1 ( F i b n + F i b n − 1 ) = F i b n 2 − F i b n + 1 F i b n − 1 = − c n c_{n+1}=\mathrm{Fib}_{n+2}\mathrm{Fib}_{n}-\mathrm{Fib}_{n+1}^{2}=(\mathrm{Fib}_{n+1}+\mathrm{Fib}_{n})\mathrm{Fib}_{n}-\mathrm{Fib}_{n+1}(\mathrm{Fib}_{n}+\mathrm{Fib}_{n-1})=\mathrm{Fib}_{n}^{2}-\mathrm{Fib}_{n+1}\mathrm{Fib}_{n-1}=-c_n c n + 1 = Fib n + 2 Fib n − Fib n + 1 2 = ( Fib n + 1 + Fib n ) Fib n − Fib n + 1 ( Fib n + Fib n − 1 ) = Fib n 2 − Fib n + 1 Fib n − 1 = − c n
Proposition 5.3. For n ≥ 2 n\ge2 n ≥ 2 x n = F i b n + 1 / F i b n x_n=\mathrm{Fib}_{n+1}/\mathrm{Fib}_{n} x n = Fib n + 1 / Fib n x n > 1 x_n>1 x n > 1 P ( x n ) = ( − 1 ) n / F i b n 2 P(x_n)=(-1)^{n}/\mathrm{Fib}_{n}^{2} P ( x n ) = ( − 1 ) n / Fib n 2
ρ − ( x n ) = ( − 1 ) n F i b n 2 u , ρ + ( x n ) = ( − 1 ) n F i b n + 1 F i b n 3 u . \rho_{-}(x_n)=\frac{(-1)^{n}}{\mathrm{Fib}_{n}^{2}}u,\qquad\rho_{+}(x_n)=\frac{(-1)^{n}\mathrm{Fib}_{n+1}}{\mathrm{Fib}_{n}^{3}}u. ρ − ( x n ) = Fib n 2 ( − 1 ) n u , ρ + ( x n ) = Fib n 3 ( − 1 ) n Fib n + 1 u . In particular x n ≠ φ x_n\ne\varphi x n = φ n n n ∣ ρ − ( x n ) ∣ = u / F i b n 2 → 0 |\rho_{-}(x_n)|=u/\mathrm{Fib}_{n}^{2}\to0 ∣ ρ − ( x n ) ∣ = u / Fib n 2 → 0
Proof. For n ≥ 2 n\ge2 n ≥ 2 F i b n − 1 ≥ 1 \mathrm{Fib}_{n-1}\ge1 Fib n − 1 ≥ 1 F i b n + 1 = F i b n + F i b n − 1 > F i b n \mathrm{Fib}_{n+1}=\mathrm{Fib}_{n}+\mathrm{Fib}_{n-1}>\mathrm{Fib}_{n} Fib n + 1 = Fib n + Fib n − 1 > Fib n F i b n 2 P ( x n ) = F i b n + 1 2 − F i b n + 1 F i b n − F i b n 2 = F i b n + 1 ( F i b n + 1 − F i b n ) − F i b n 2 = F i b n + 1 F i b n − 1 − F i b n 2 = ( − 1 ) n \mathrm{Fib}_{n}^{2}P(x_n)=\mathrm{Fib}_{n+1}^{2}-\mathrm{Fib}_{n+1}\mathrm{Fib}_{n}-\mathrm{Fib}_{n}^{2}=\mathrm{Fib}_{n+1}(\mathrm{Fib}_{n+1}-\mathrm{Fib}_{n})-\mathrm{Fib}_{n}^{2}=\mathrm{Fib}_{n+1}\mathrm{Fib}_{n-1}-\mathrm{Fib}_{n}^{2}=(-1)^{n} Fib n 2 P ( x n ) = Fib n + 1 2 − Fib n + 1 Fib n − Fib n 2 = Fib n + 1 ( Fib n + 1 − Fib n ) − Fib n 2 = Fib n + 1 Fib n − 1 − Fib n 2 = ( − 1 ) n Definition 5.1 .
Remark 5.4 (the integer shadow). For n ≥ 3 n\ge3 n ≥ 3 N > F i b n + 2 N>\mathrm{Fib}_{n+2} N > Fib n + 2 Lemma 3.1 with α = F i b n \alpha=\mathrm{Fib}_{n} α = Fib n β = F i b n + 1 \beta=\mathrm{Fib}_{n+1} β = Fib n + 1 ( N , N − F i b n , N − F i b n + 1 , N − F i b n + 2 ) (N,N-\mathrm{Fib}_{n},N-\mathrm{Fib}_{n+1},N-\mathrm{Fib}_{n+2}) ( N , N − Fib n , N − Fib n + 1 , N − Fib n + 2 ) F i b n − 1 , F i b n , F i b n + 1 , F i b n + 1 , F i b n + 2 \mathrm{Fib}_{n-1},\mathrm{Fib}_{n},\mathrm{Fib}_{n+1},\mathrm{Fib}_{n+1},\mathrm{Fib}_{n+2} Fib n − 1 , Fib n , Fib n + 1 , Fib n + 1 , Fib n + 2 1 , 2 , 2 , 1 1,2,2,1 1 , 2 , 2 , 1 n ≥ 3 n\ge3 n ≥ 3 ( N , N − 2 , N − 3 , N − 5 ) (N,N-2,N-3,N-5) ( N , N − 2 , N − 3 , N − 5 ) 1 , 2 , 2 , 3 , 3 , 5 1,2,2,3,3,5 1 , 2 , 2 , 3 , 3 , 5 β − α = F i b n − 1 \beta-\alpha=\mathrm{Fib}_{n-1} β − α = Fib n − 1 α + β = F i b n + 2 \alpha+\beta=\mathrm{Fib}_{n+2} α + β = Fib n + 2 ( c x k ) (cx^{k}) ( c x k ) c > 0 c>0 c > 0 x > 1 x>1 x > 1 c x k + 2 = c x k + 1 + c x k cx^{k+2}=cx^{k+1}+cx^{k} c x k + 2 = c x k + 1 + c x k x = φ x=\varphi x = φ
Rigidity for arbitrary four-point sets
Theorem 3 concerns quadruples of the special form S x S_x S x
Theorem 3′ (Rigidity). Let x > 1 x > 1 x > 1 c > 0 c > 0 c > 0 T ⊂ R T \subset\mathbb{R} T ⊂ R
Δ ( T ) = V : = { c , c x , c x 2 , c x 3 } \Delta(T) = V := \{c,\; cx,\; cx^2,\; cx^3\} Δ ( T ) = V := { c , c x , c x 2 , c x 3 } if and only if x = φ x = \varphi x = φ T = t + c { 0 , φ , φ 2 , φ 3 } T = t + c\{0,\varphi,\varphi^2,\varphi^3\} T = t + c { 0 , φ , φ 2 , φ 3 } t ∈ R t \in\mathbb{R} t ∈ R T T T S ( F , c ) S(F,c) S ( F , c ) r T ( c ) = r T ( c φ 3 ) = 1 r_T(c) = r_T(c\varphi^3) = 1 r T ( c ) = r T ( c φ 3 ) = 1 r T ( c φ ) = r T ( c φ 2 ) = 2 r_T(c\varphi) = r_T(c\varphi^2) = 2 r T ( c φ ) = r T ( c φ 2 ) = 2
Proof. “If” is Theorem 1 with u = c u = c u = c T = { y 1 < y 2 < y 3 < y 4 } T = \{y_1 < y_2 < y_3 < y_4\} T = { y 1 < y 2 < y 3 < y 4 } Δ ( T ) = V \Delta(T) = V Δ ( T ) = V g 1 , g 2 , g 3 g_1,g_2,g_3 g 1 , g 2 , g 3 σ 1 = g 1 + g 2 = y 3 − y 1 \sigma_1 = g_1 + g_2 = y_3 - y_1 σ 1 = g 1 + g 2 = y 3 − y 1 σ 2 = g 2 + g 3 = y 4 − y 2 \sigma_2 = g_2 + g_3 = y_4 - y_2 σ 2 = g 2 + g 3 = y 4 − y 2 y 4 − y 1 y_4 - y_1 y 4 − y 1 max V = c x 3 \max V = cx^3 max V = c x 3 g 1 , g 2 , g 3 , σ 1 , σ 2 g_1,g_2,g_3,\sigma_1,\sigma_2 g 1 , g 2 , g 3 , σ 1 , σ 2 V V V ≥ min V = c \ge\min V = c ≥ min V = c σ 1 , σ 2 ≥ 2 c > c \sigma_1,\sigma_2 \ge2c > c σ 1 , σ 2 ≥ 2 c > c σ 1 , σ 2 < c x 3 \sigma_1,\sigma_2 < cx^3 σ 1 , σ 2 < c x 3 σ 1 , σ 2 ∈ { c x , c x 2 } \sigma_1,\sigma_2 \in\{cx,cx^2\} σ 1 , σ 2 ∈ { c x , c x 2 }
Step 1: σ 1 = σ 2 = c x 2 \sigma_1 = \sigma_2 = cx^2 σ 1 = σ 2 = c x 2 σ 1 = c x \sigma_1 = cx σ 1 = c x g 1 , g 2 ∈ V g_1,g_2 \in V g 1 , g 2 ∈ V g 1 , g 2 < σ 1 = c x g_1,g_2 < \sigma_1 = cx g 1 , g 2 < σ 1 = c x V V V c x cx c x c c c g 1 = g 2 = c g_1 = g_2 = c g 1 = g 2 = c x = 2 x = 2 x = 2 g 3 = c x 3 − σ 1 = 6 c ∉ { c , 2 c , 4 c , 8 c } = V g_3 = cx^3 - \sigma_1 = 6c \notin\{c,2c,4c,8c\} = V g 3 = c x 3 − σ 1 = 6 c ∈ / { c , 2 c , 4 c , 8 c } = V σ 2 = c x \sigma_2 = cx σ 2 = c x g 2 = g 3 = c g_2 = g_3 = c g 2 = g 3 = c x = 2 x = 2 x = 2 g 1 = 6 c ∉ V g_1 = 6c \notin V g 1 = 6 c ∈ / V
Step 2: Now g 3 = c x 3 − σ 1 = c x 2 ( x − 1 ) g_3 = cx^3 - \sigma_1 = cx^2(x - 1) g 3 = c x 3 − σ 1 = c x 2 ( x − 1 ) g 1 = c x 3 − σ 2 = c x 2 ( x − 1 ) g_1 = cx^3 - \sigma_2 = cx^2(x - 1) g 1 = c x 3 − σ 2 = c x 2 ( x − 1 ) g 2 = σ 1 − g 1 = c x 2 ( 2 − x ) g_2 = \sigma_1 - g_1 = cx^2(2 - x) g 2 = σ 1 − g 1 = c x 2 ( 2 − x ) g 2 > 0 g_2 > 0 g 2 > 0 x < 2 x < 2 x < 2 g 3 ∈ V g_3 \in V g 3 ∈ V x 2 ( x − 1 ) ∈ { 1 , x , x 2 , x 3 } x^2(x - 1) \in\{1,x,x^2,x^3\} x 2 ( x − 1 ) ∈ { 1 , x , x 2 , x 3 } x 3 x^3 x 3 x 2 x^2 x 2 x = 2 x = 2 x = 2 x x x x 2 − x − 1 = 0 x^2 - x - 1 = 0 x 2 − x − 1 = 0 x = φ x = \varphi x = φ 1 1 1 x 3 − x 2 = 1 x^3 - x^2 = 1 x 3 − x 2 = 1
Step 3: x 3 − x 2 = 1 x^3 - x^2 = 1 x 3 − x 2 = 1 g 1 = g 3 = c g_1 = g_3 = c g 1 = g 3 = c g 2 = c ( 2 x 2 − x 3 ) = c ( x 2 − 1 ) g_2 = c(2x^2 - x^3) = c(x^2 - 1) g 2 = c ( 2 x 2 − x 3 ) = c ( x 2 − 1 ) g 2 ∈ V g_2 \in V g 2 ∈ V x 2 − 1 ∈ { 1 , x , x 2 , x 3 } x^2 - 1 \in\{1,x,x^2,x^3\} x 2 − 1 ∈ { 1 , x , x 2 , x 3 } x 2 − 1 = 1 x^2 - 1 = 1 x 2 − 1 = 1 x = 2 x = \sqrt{2} x = 2 ( 2 ) 3 − ( 2 ) 2 = 2 2 − 2 ≠ 1 (\sqrt{2})^3 - (\sqrt{2})^2 = 2\sqrt{2} - 2 \ne1 ( 2 ) 3 − ( 2 ) 2 = 2 2 − 2 = 1 8 ≠ 9 8 \ne9 8 = 9 x 2 − 1 = x x^2 - 1 = x x 2 − 1 = x x = φ x = \varphi x = φ φ 3 − φ 2 = φ ≠ 1 \varphi^3 - \varphi^2 = \varphi\ne1 φ 3 − φ 2 = φ = 1 x 2 − 1 = x 2 x^2 - 1 = x^2 x 2 − 1 = x 2 x 2 − 1 = x 3 x^2 - 1 = x^3 x 2 − 1 = x 3 x 3 − x 2 = − 1 ≠ 1 x^3 - x^2 = -1 \ne1 x 3 − x 2 = − 1 = 1
Step 4: x = φ x = \varphi x = φ g 1 = g 3 = c φ 2 ( φ − 1 ) = c φ g_1 = g_3 = c\varphi^2(\varphi- 1) = c\varphi g 1 = g 3 = c φ 2 ( φ − 1 ) = c φ g 2 = c φ 2 ( 2 − φ ) = c g_2 = c\varphi^2(2 - \varphi) = c g 2 = c φ 2 ( 2 − φ ) = c T = { y 1 , y 1 + c φ , y 1 + c φ 2 , y 1 + c φ 3 } T = \{y_1, y_1 + c\varphi, y_1 + c\varphi^2, y_1 + c\varphi^3\} T = { y 1 , y 1 + c φ , y 1 + c φ 2 , y 1 + c φ 3 }
Remark 5.5. The proof shows that apart from x = φ x = \varphi x = φ x = 2 x = 2 x = 2 x ≈ 1.46557 x \approx1.46557 x ≈ 1.46557 x 3 = x 2 + 1 x^3 = x^2 + 1 x 3 = x 2 + 1 x = 2 x = 2 x = 2 g 3 = 6 c ∉ V g_3 = 6c \notin V g 3 = 6 c ∈ / V g 2 = c x 2 ( 2 − x ) = 0 g_2 = cx^2(2 - x) = 0 g 2 = c x 2 ( 2 − x ) = 0 x 3 = x 2 + 1 x^3 = x^2 + 1 x 3 = x 2 + 1 g 2 = c ( x 2 − 1 ) g_2 = c(x^2 - 1) g 2 = c ( x 2 − 1 ) V V V 0 < x < 1 0 < x < 1 0 < x < 1 x x x 1 / x 1/x 1/ x x ≠ 1 x \ne1 x = 1 x ∈ { φ , φ − 1 } x \in\{\varphi,\varphi^{-1}\} x ∈ { φ , φ − 1 } { y 1 < y 2 < y 3 } \{y_1 < y_2 < y_3\} { y 1 < y 2 < y 3 } Δ = { c , c x , c x 2 } \Delta= \{c,cx,cx^2\} Δ = { c , c x , c x 2 } x > 1 x > 1 x > 1 c x 2 cx^2 c x 2 { g 1 , g 2 } = { c , c x } \{g_1,g_2\} = \{c,cx\} { g 1 , g 2 } = { c , c x } c + c x = c x 2 c + cx = cx^2 c + c x = c x 2 x = φ x = \varphi x = φ
Remark 5.6 (Relation to [13 Lemma 3] ). The proof of Theorem 3′ is self-contained and does not use [13 ] . Each three-point subset of a set T T T x l = x k + 1 x^l = x^k + 1 x l = x k + 1 l > k ≥ 0 l > k \ge0 l > k ≥ 0 x x x X l − X k − 1 X^l - X^k - 1 X l − X k − 1 [13 Lemma 3] (with a computer-assisted step), the common roots x ≥ 1.05 x \ge1.05 x ≥ 1.05 φ 1 / m \varphi^{1/m} φ 1/ m m = 1 , … , 5 m = 1,\ldots,5 m = 1 , … , 5 φ \varphi φ X 2 − X − 1 X^2 - X - 1 X 2 − X − 1 X l − X k − 1 X^l - X^k - 1 X l − X k − 1 l > k ≥ 0 l > k \ge0 l > k ≥ 0 φ \varphi φ { 0 , φ , φ 2 , φ 3 } \{0,\varphi,\varphi^2,\varphi^3\} { 0 , φ , φ 2 , φ 3 } [13 Lemma 3] and needs no computation. Theorem 3′ can also be derived from that lemma by a short argument. The span c x 3 cx^3 c x 3 T T T ≥ c \ge c ≥ c x ≥ 3 3 > φ x \ge\sqrt[3]{3} > \varphi x ≥ 3 3 > φ x x x X l − X k − 1 X^l - X^k - 1 X l − X k − 1 x x x x ≥ 3 3 > 2 x \ge\sqrt[3]{3} > \sqrt{2} x ≥ 3 3 > 2 3 2 > 2 3 3^2 > 2^3 3 2 > 2 3 x l − k = 1 + x − k ≤ 2 < x 2 x^{l-k} = 1 + x^{-k} \le2 < x^2 x l − k = 1 + x − k ≤ 2 < x 2 l = k + 1 l = k + 1 l = k + 1 x k ( x − 1 ) = 1 x^k(x - 1) = 1 x k ( x − 1 ) = 1 k k k x k ( x − 1 ) x^k(x - 1) x k ( x − 1 ) k k k [13 Lemma 3] is used only for the uniqueness of the trinomial, which here is elementary, so all three-point subsets of T T T T = { 0 , p , q , 1 } T = \{0,p,q,1\} T = { 0 , p , q , 1 } { 0 , p , 1 } \{0,p,1\} { 0 , p , 1 } { 0 , q , 1 } \{0,q,1\} { 0 , q , 1 } q = 1 − p q = 1 - p q = 1 − p 0 < p < 1 2 0 < p < \frac{1}{2} 0 < p < 2 1 { 0 , p , 1 − p } \{0,p,1-p\} { 0 , p , 1 − p } p / ( 1 − p ) p/(1-p) p / ( 1 − p ) ( 1 − 2 p ) / ( 1 − p ) (1-2p)/(1-p) ( 1 − 2 p ) / ( 1 − p ) p / ( 1 − p ) = 1 − p p/(1-p) = 1-p p / ( 1 − p ) = 1 − p p / ( 1 − p ) = p p/(1-p) = p p / ( 1 − p ) = p p = 0 p = 0 p = 0 p = ( 1 − p ) 2 p = (1-p)^2 p = ( 1 − p ) 2 p = φ − 2 p = \varphi^{-2} p = φ − 2 φ − 3 , φ − 2 , φ − 1 , 1 \varphi^{-3},\varphi^{-2},\varphi^{-1},1 φ − 3 , φ − 2 , φ − 1 , 1 x = φ x = \varphi x = φ
The cosine sum: non-periodic, but almost periodic
Lemma 6.1. For finite A ⊂ R A \subset\mathbb{R} A ⊂ R t ∈ R t \in\mathbb{R} t ∈ R ∣ A ^ ( t ) ∣ 2 = ∣ A ∣ + 2 C A ( t ) |\widehat{A}(t)|^2 = |A| + 2C_A(t) ∣ A ( t ) ∣ 2 = ∣ A ∣ + 2 C A ( t ) − ∣ A ∣ / 2 ≤ C A ( t ) ≤ ( ∣ A ∣ 2 ) -|A|/2 \le C_A(t) \le\binom{|A|}{2} − ∣ A ∣/2 ≤ C A ( t ) ≤ ( 2 ∣ A ∣ ) C A ( t ) = ( ∣ A ∣ 2 ) C_A(t) = \binom{|A|}{2} C A ( t ) = ( 2 ∣ A ∣ ) d t ∈ Z dt \in\mathbb{Z} d t ∈ Z d ∈ Δ ( A ) d \in\Delta(A) d ∈ Δ ( A )
Proof. ∣ A ^ ( t ) ∣ 2 = ∑ x , y ∈ A e ( ( x − y ) t ) = ∣ A ∣ + ∑ x > y ( e ( ( x − y ) t ) + e ( − ( x − y ) t ) ) = ∣ A ∣ + 2 C A ( t ) |\widehat{A}(t)|^2 = \sum_{x,y\in A}e((x-y)t)=|A|+\sum_{x>y}(e((x-y)t)+e(-(x-y)t))=|A|+2C_A(t) ∣ A ( t ) ∣ 2 = ∑ x , y ∈ A e (( x − y ) t ) = ∣ A ∣ + ∑ x > y ( e (( x − y ) t ) + e ( − ( x − y ) t )) = ∣ A ∣ + 2 C A ( t ) ∣ A ^ ∣ 2 ≥ 0 |\widehat{A}|^2\ge0 ∣ A ∣ 2 ≥ 0 C A C_A C A ( ∣ A ∣ 2 ) \binom{|A|}{2} ( 2 ∣ A ∣ ) 1 1 1 ( x − y ) t ∈ Z (x-y)t\in\mathbb{Z} ( x − y ) t ∈ Z x > y x>y x > y A A A
Proposition 6.2 (closed form). For all t ∈ R t\in\mathbb{R} t ∈ R
C S ( t ) = cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t ) = 8 cos 2 ( π a t ) cos 2 ( π b t ) − 2. C_S(t)=\cos(2\pi ut)+2\cos(2\pi\varphi ut)+2\cos(2\pi\varphi^2ut)+\cos(2\pi\varphi^3ut)=8\cos^2(\pi at)\cos^2(\pi bt)-2. C S ( t ) = cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t ) = 8 cos 2 ( π a t ) cos 2 ( π b t ) − 2. Equivalently, for the exponential sum S ^ ( t ) = ∑ x ∈ S e ( x t ) \widehat{S}(t)=\sum_{x\in S}e(xt) S ( t ) = ∑ x ∈ S e ( x t ) S S S ∣ S ^ ( t ) ∣ 2 = 4 + 2 C S ( t ) = 16 cos 2 ( π a t ) cos 2 ( π b t ) |\widehat{S}(t)|^2=4+2C_S(t)=16\cos^2(\pi at)\cos^2(\pi bt) ∣ S ( t ) ∣ 2 = 4 + 2 C S ( t ) = 16 cos 2 ( π a t ) cos 2 ( π b t ) ∣ S ^ ( t ) ∣ = 4 ∣ cos ( π a t ) cos ( π b t ) ∣ |\widehat{S}(t)|=4|\cos(\pi at)\cos(\pi bt)| ∣ S ( t ) ∣ = 4∣ cos ( π a t ) cos ( π b t ) ∣
− 2 ≤ C S ( t ) ≤ 6 -2\le C_S(t)\le6 − 2 ≤ C S ( t ) ≤ 6 t t t
C S ( t ) = 6 C_S(t)=6 C S ( t ) = 6 t = 0 t=0 t = 0
C S ( t ) = − 2 C_S(t)=-2 C S ( t ) = − 2 t ∈ 2 Z + 1 2 a ∪ 2 Z + 1 2 b t\in\frac{2\mathbb{Z}+1}{2a}\cup\frac{2\mathbb{Z}+1}{2b} t ∈ 2 a 2 Z + 1 ∪ 2 b 2 Z + 1
Proof. The first expression is Definition 2.10 combined with Theorem 1. Since S = F − ( { 0 , a } + { 0 , b } ) S=F-\left(\{0,a\}+\{0,b\}\right) S = F − ( { 0 , a } + { 0 , b } )
S ^ ( t ) = e ( F t ) ∑ q ∈ { 0 , a , b , a + b } e ( − q t ) = e ( F t ) ( 1 + e ( − a t ) ) ( 1 + e ( − b t ) ) . \widehat{S}(t)=e(Ft)\sum_{q\in\{0,a,b,a+b\}}e(-qt)=e(Ft)(1+e(-at))(1+e(-bt)). S ( t ) = e ( F t ) q ∈ { 0 , a , b , a + b } ∑ e ( − q t ) = e ( F t ) ( 1 + e ( − a t )) ( 1 + e ( − b t )) . As ∣ 1 + e ( θ ) ∣ 2 = 2 + 2 cos 2 π θ = 4 cos 2 π θ |1+e(\theta)|^2=2+2\cos2\pi\theta=4\cos^2\pi\theta ∣1 + e ( θ ) ∣ 2 = 2 + 2 cos 2 π θ = 4 cos 2 π θ ∣ S ^ ( t ) ∣ 2 = 16 cos 2 ( π a t ) cos 2 ( π b t ) |\widehat{S}(t)|^2=16\cos^2(\pi at)\cos^2(\pi bt) ∣ S ( t ) ∣ 2 = 16 cos 2 ( π a t ) cos 2 ( π b t ) Lemma 6.1 with ∣ S ∣ = 4 |S|=4 ∣ S ∣ = 4 C S = ( ∣ S ^ ∣ 2 − 4 ) / 2 C_S=(|\widehat{S}|^2-4)/2 C S = ( ∣ S ∣ 2 − 4 ) /2 0 ≤ cos 2 ≤ 1 0\le\cos^2\le1 0 ≤ cos 2 ≤ 1 C S ( t ) = 6 C_S(t)=6 C S ( t ) = 6 cos 2 ( π a t ) = cos 2 ( π b t ) = 1 \cos^2(\pi at)=\cos^2(\pi bt)=1 cos 2 ( π a t ) = cos 2 ( π b t ) = 1 a t ∈ Z at\in\mathbb{Z} a t ∈ Z b t ∈ Z bt\in\mathbb{Z} b t ∈ Z t ≠ 0 t\ne0 t = 0 a t ≠ 0 at\ne0 a t = 0 b / a = ( b t ) / ( a t ) ∈ Q b/a=(bt)/(at)\in\mathbb{Q} b / a = ( b t ) / ( a t ) ∈ Q b / a = φ ∉ Q b/a=\varphi\notin\mathbb{Q} b / a = φ ∈ / Q Lemma 2.4 (iv)). (iii) C S ( t ) = − 2 C_S(t)=-2 C S ( t ) = − 2 cos ( π a t ) = 0 \cos(\pi at)=0 cos ( π a t ) = 0 cos ( π b t ) = 0 \cos(\pi bt)=0 cos ( π b t ) = 0 a t ∈ 1 2 + Z at\in\frac{1}{2}+\mathbb{Z} a t ∈ 2 1 + Z b t ∈ 1 2 + Z bt\in\frac{1}{2}+\mathbb{Z} b t ∈ 2 1 + Z
Lemma 6.3 (trigonometric polynomials are almost periodic). Let λ 1 , … , λ k ∈ R \lambda_1,\ldots,\lambda_k\in\mathbb{R} λ 1 , … , λ k ∈ R c 1 , … , c k ∈ C c_1,\ldots,c_k\in\mathbb{C} c 1 , … , c k ∈ C P ( t ) = ∑ j = 1 k c j e ( λ j t ) P(t)=\sum_{j=1}^k c_je(\lambda_jt) P ( t ) = ∑ j = 1 k c j e ( λ j t )
(a) For all t , τ ∈ R t,\tau\in\mathbb{R} t , τ ∈ R ∣ P ( t + τ ) − P ( t ) ∣ ≤ 2 π ∑ j ∣ c j ∣ ∥ λ j τ ∥ |P(t+\tau)-P(t)|\le2\pi\sum_j|c_j|\|\lambda_j\tau\| ∣ P ( t + τ ) − P ( t ) ∣ ≤ 2 π ∑ j ∣ c j ∣∥ λ j τ ∥ ∣ cos ( 2 π λ ( t + τ ) ) − cos ( 2 π λ t ) ∣ ≤ 2 π ∥ λ τ ∥ |\cos(2\pi\lambda(t+\tau))-\cos(2\pi\lambda t)|\le2\pi\|\lambda\tau\| ∣ cos ( 2 π λ ( t + τ )) − cos ( 2 π λ t ) ∣ ≤ 2 π ∥ λ τ ∥ λ \lambda λ
(b) For every δ > 0 \delta>0 δ > 0 B ( δ ) = { τ ∈ R : ∥ λ j τ ∥ < δ for all j } B(\delta)=\{\tau\in\mathbb{R}:\|\lambda_j\tau\|<\delta\text{ for all }j\} B ( δ ) = { τ ∈ R : ∥ λ j τ ∥ < δ for all j }
(c) P P P t ↦ cos ( 2 π λ t ) t\mapsto\cos(2\pi\lambda t) t ↦ cos ( 2 π λ t )
Proof. (a) P ( t + τ ) − P ( t ) = ∑ j c j e ( λ j t ) ( e ( λ j τ ) − 1 ) P(t+\tau)-P(t)=\sum_j c_je(\lambda_jt)(e(\lambda_j\tau)-1) P ( t + τ ) − P ( t ) = ∑ j c j e ( λ j t ) ( e ( λ j τ ) − 1 ) Lemma 2.3 . For the cosine, ∣ cos ( 2 π λ ( t + τ ) ) − cos ( 2 π λ t ) ∣ = ∣ Re ( e ( λ t ) ( e ( λ τ ) − 1 ) ) ∣ ≤ ∣ e ( λ τ ) − 1 ∣ |\cos(2\pi\lambda(t+\tau))-\cos(2\pi\lambda t)|=|\operatorname{Re}(e(\lambda t)(e(\lambda\tau)-1))|\le|e(\lambda\tau)-1| ∣ cos ( 2 π λ ( t + τ )) − cos ( 2 π λ t ) ∣ = ∣ Re ( e ( λ t ) ( e ( λ τ ) − 1 )) ∣ ≤ ∣ e ( λ τ ) − 1∣
(b) Let T k = R k / Z k \mathbb{T}^k=\mathbb{R}^k/\mathbb{Z}^k T k = R k / Z k π \pi π d ( π ( ξ ) , π ( η ) ) = max j ∥ ξ j − η j ∥ d(\pi(\xi),\pi(\eta))=\max_j\|\xi_j-\eta_j\| d ( π ( ξ ) , π ( η )) = max j ∥ ξ j − η j ∥ Lemma 2.3 this is a well-defined, translation-invariant metric on T k \mathbb{T}^k T k d ( π ( ξ ) , π ( η ) ) ≤ max j ∣ ξ j − η j ∣ d(\pi(\xi),\pi(\eta))\le\max_j|\xi_j-\eta_j| d ( π ( ξ ) , π ( η )) ≤ max j ∣ ξ j − η j ∣ π \pi π T k = π ( [ 0 , 1 ] k ) \mathbb{T}^k=\pi([0,1]^k) T k = π ([ 0 , 1 ] k ) Θ : R → T k \Theta:\mathbb{R}\to\mathbb{T}^k Θ : R → T k Θ ( s ) = π ( λ 1 s , … , λ k s ) \Theta(s)=\pi(\lambda_1s,\ldots,\lambda_ks) Θ ( s ) = π ( λ 1 s , … , λ k s ) K K K Θ ( R ) \Theta(\mathbb{R}) Θ ( R ) δ \delta δ Θ ( s ) \Theta(s) Θ ( s ) s ∈ R s\in\mathbb{R} s ∈ R K K K Θ ( s 1 ) , … , Θ ( s r ) \Theta(s_1),\ldots,\Theta(s_r) Θ ( s 1 ) , … , Θ ( s r ) K K K R = max i ∣ s i ∣ R=\max_i|s_i| R = max i ∣ s i ∣ σ ∈ R \sigma\in\mathbb{R} σ ∈ R Θ ( − σ ) ∈ K \Theta(-\sigma)\in K Θ ( − σ ) ∈ K d ( Θ ( − σ ) , Θ ( s i ) ) < δ d(\Theta(-\sigma),\Theta(s_i))<\delta d ( Θ ( − σ ) , Θ ( s i )) < δ Θ ( σ ) \Theta(\sigma) Θ ( σ ) d ( 0 , Θ ( σ + s i ) ) < δ d(0,\Theta(\sigma+s_i))<\delta d ( 0 , Θ ( σ + s i )) < δ τ = σ + s i ∈ B ( δ ) \tau=\sigma+s_i\in B(\delta) τ = σ + s i ∈ B ( δ ) ∣ τ − σ ∣ ≤ R |\tau-\sigma|\le R ∣ τ − σ ∣ ≤ R L = 2 R + 1 L=2R+1 L = 2 R + 1 B ( δ ) B(\delta) B ( δ )
(c) P P P ε > 0 \varepsilon>0 ε > 0 δ = ε 2 π ( 1 + ∑ j ∣ c j ∣ ) \delta=\frac{\varepsilon}{2\pi(1+\sum_j|c_j|)} δ = 2 π ( 1 + ∑ j ∣ c j ∣ ) ε B ( δ ) ⊆ E ( P ; ε ) B(\delta)\subseteq E(P;\varepsilon) B ( δ ) ⊆ E ( P ; ε ) cos ( 2 π λ t ) = 1 2 e ( λ t ) + 1 2 e ( − λ t ) \cos(2\pi\lambda t)=\frac{1}{2}e(\lambda t)+\frac{1}{2}e(-\lambda t) cos ( 2 π λ t ) = 2 1 e ( λ t ) + 2 1 e ( − λ t )
Proposition 6.4 (periodicity criterion). Let A ⊂ R A\subset\mathbb{R} A ⊂ R ∣ A ∣ ≥ 2 |A|\ge2 ∣ A ∣ ≥ 2 C A ( t ) = ∑ d ∈ Δ ( A ) r A ( d ) cos ( 2 π d t ) C_A(t)=\sum_{d\in\Delta(A)}r^A(d)\cos(2\pi dt) C A ( t ) = ∑ d ∈ Δ ( A ) r A ( d ) cos ( 2 π d t ) d d d d / d ′ ∈ Q d/d'\in\mathbb{Q} d / d ′ ∈ Q d , d ′ ∈ Δ ( A ) d,d'\in\Delta(A) d , d ′ ∈ Δ ( A )
This criterion is well known. It is a special case of a theorem of Mirotin and Mirotin [34 §2] : a sum of continuous, nonconstant periodic functions on R \mathbb{R} R r A ( d ) cos ( 2 π d t ) r_A(d)\cos(2\pi dt) r A ( d ) cos ( 2 π d t ) 1 / d 1/d 1/ d 1 / d 1/d 1/ d 1 / d ′ 1/d' 1/ d ′ d / d ′ ∈ Q d/d'\in\mathbb{Q} d / d ′ ∈ Q ∑ d ∈ E r A ( d ) cos ( 2 π d t ) \sum_{d\in E}r_A(d)\cos(2\pi dt) ∑ d ∈ E r A ( d ) cos ( 2 π d t ) ∅ ≠ E ⊆ Δ ( A ) \varnothing\ne E\subseteq\Delta(A) ∅ = E ⊆ Δ ( A ) t = 0 t=0 t = 0 ∑ d ∈ E r A ( d ) > 0 \sum_{d\in E}r_A(d)>0 ∑ d ∈ E r A ( d ) > 0 1 T ∫ 0 T \frac{1}{T}\int_0^T T 1 ∫ 0 T 0 0 0 T → ∞ T\to\infty T → ∞
Proof. (⇐ \Leftarrow ⇐ d 0 ∈ Δ ( A ) d_0\in\Delta(A) d 0 ∈ Δ ( A ) d = ( p d / q d ) d 0 d=(p_d/q_d)d_0 d = ( p d / q d ) d 0 p d , q d p_d,q_d p d , q d Q = lcm d q d Q=\operatorname{lcm}_d q_d Q = lcm d q d T = Q / d 0 T=Q/d_0 T = Q / d 0 d T = p d Q / q d ∈ Z dT=p_dQ/q_d\in\mathbb{Z} d T = p d Q / q d ∈ Z cos ( 2 π d t ) \cos(2\pi dt) cos ( 2 π d t ) T T T ⇒ \Rightarrow ⇒ T > 0 T>0 T > 0 C A ( T ) = C A ( 0 ) = ( ∣ A ∣ 2 ) C_A(T)=C_A(0)=\binom{|A|}{2} C A ( T ) = C A ( 0 ) = ( 2 ∣ A ∣ ) d T ∈ Z ∖ { 0 } dT\in\mathbb{Z}\setminus\{0\} d T ∈ Z ∖ { 0 } d ∈ Δ ( A ) d\in\Delta(A) d ∈ Δ ( A ) Lemma 6.1 , and d / d ′ = ( d T ) / ( d ′ T ) ∈ Q d/d'=(dT)/(d'T)\in\mathbb{Q} d / d ′ = ( d T ) / ( d ′ T ) ∈ Q
Theorem 4. Let C S ( t ) = ∑ 1 ≤ i < j ≤ 4 cos ( 2 π ( s i − s j ) t ) = cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t ) C_S(t)=\sum_{1\le i<j\le4}\cos(2\pi(s_i-s_j)t)=\cos(2\pi ut)+2\cos(2\pi\varphi ut)+2\cos(2\pi\varphi^2ut)+\cos(2\pi\varphi^3ut) C S ( t ) = ∑ 1 ≤ i < j ≤ 4 cos ( 2 π ( s i − s j ) t ) = cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t )
(a) C S C_S C S
(b) C S C_S C S t , τ ∈ R t,\tau\in\mathbb{R} t , τ ∈ R
∣ C S ( t + τ ) − C S ( t ) ∣ ≤ 2 π ( 4 ∥ u τ ∥ + 6 ∥ φ u τ ∥ ) , |C_S(t+\tau)-C_S(t)|\le2\pi(4\lVert u\tau\rVert+6\lVert\varphi u\tau\rVert), ∣ C S ( t + τ ) − C S ( t ) ∣ ≤ 2 π ( 4 ∥ uτ ∥ + 6 ∥ φ uτ ∥) , so E ( C S ; ε ) E(C_S;\varepsilon) E ( C S ; ε ) { τ : ∥ u τ ∥ < ε / ( 20 π ) , ∥ φ u τ ∥ < ε / ( 20 π ) } \{\tau:\lVert u\tau\rVert<\varepsilon/(20\pi),\ \lVert\varphi u\tau\rVert<\varepsilon/(20\pi)\} { τ : ∥ uτ ∥ < ε / ( 20 π ) , ∥ φ uτ ∥ < ε / ( 20 π )}
Proof. (a) If T > 0 T>0 T > 0 C S ( T ) = C S ( 0 ) = 6 C_S(T)=C_S(0)=6 C S ( T ) = C S ( 0 ) = 6 Proposition 6.2 (ii). (Alternatively: u u u φ u \varphi u φ u Proposition 6.4 applies.)
(b) By Lemma 6.3 (a), ∣ C S ( t + τ ) − C S ( t ) ∣ ≤ 2 π ∑ d r S ( d ) ∥ d τ ∥ |C_S(t+\tau)-C_S(t)|\le2\pi\sum_d r_S(d)\lVert d\tau\rVert ∣ C S ( t + τ ) − C S ( t ) ∣ ≤ 2 π ∑ d r S ( d ) ∥ d τ ∥ θ 1 = u τ \theta_1=u\tau θ 1 = uτ θ 2 = φ u τ \theta_2=\varphi u\tau θ 2 = φ uτ φ 2 u τ = θ 1 + θ 2 \varphi^2u\tau=\theta_1+\theta_2 φ 2 uτ = θ 1 + θ 2 φ 3 u τ = θ 1 + 2 θ 2 \varphi^3u\tau=\theta_1+2\theta_2 φ 3 uτ = θ 1 + 2 θ 2 Lemma 2.3
∑ d r S ( d ) ∥ d τ ∥ ≤ ∥ θ 1 ∥ + 2 ∥ θ 2 ∥ + 2 ( ∥ θ 1 ∥ + ∥ θ 2 ∥ ) + ( ∥ θ 1 ∥ + 2 ∥ θ 2 ∥ ) = 4 ∥ θ 1 ∥ + 6 ∥ θ 2 ∥ . \sum_d r_S(d)\lVert d\tau\rVert\le\lVert\theta_1\rVert+2\lVert\theta_2\rVert+2(\lVert\theta_1\rVert+\lVert\theta_2\rVert)+(\lVert\theta_1\rVert+2\lVert\theta_2\rVert)=4\lVert\theta_1\rVert+6\lVert\theta_2\rVert. d ∑ r S ( d ) ∥ d τ ∥ ≤ ∥ θ 1 ∥ + 2 ∥ θ 2 ∥ + 2 (∥ θ 1 ∥ + ∥ θ 2 ∥) + (∥ θ 1 ∥ + 2 ∥ θ 2 ∥) = 4 ∥ θ 1 ∥ + 6 ∥ θ 2 ∥ . If ∥ θ 1 ∥ , ∥ θ 2 ∥ < ε / ( 20 π ) \lVert\theta_1\rVert,\lVert\theta_2\rVert<\varepsilon/(20\pi) ∥ θ 1 ∥ , ∥ θ 2 ∥ < ε / ( 20 π ) < ε <\varepsilon < ε τ \tau τ B ( ε / ( 20 π ) ) B(\varepsilon/(20\pi)) B ( ε / ( 20 π )) ( λ 1 , λ 2 ) = ( u , φ u ) (\lambda_1,\lambda_2)=(u,\varphi u) ( λ 1 , λ 2 ) = ( u , φ u ) Lemma 6.3 (b). Since C S C_S C S
Proposition 6.5 (explicit almost periods). For n ∈ N n\in\mathbb{N} n ∈ N τ n = Fib n / u \tau_n=\operatorname{Fib}_n/u τ n = Fib n / u
(i) Fib n + 1 − φ Fib n = ( − φ − 1 ) n \operatorname{Fib}_{n+1}-\varphi\operatorname{Fib}_n=(-\varphi^{-1})^n Fib n + 1 − φ Fib n = ( − φ − 1 ) n ∥ φ Fib n ∥ ≤ φ − n \lVert\varphi\operatorname{Fib}_n\rVert\le\varphi^{-n} ∥ φ Fib n ∥ ≤ φ − n n ≥ 2 n\ge2 n ≥ 2 n = 1 n=1 n = 1 ∥ φ ∥ = φ − 2 \lVert\varphi\rVert=\varphi^{-2} ∥ φ ∥ = φ − 2
(ii) sup t ∈ R ∣ C S ( t + τ n ) − C S ( t ) ∣ ≤ 12 π φ − n \sup_{t\in\mathbb{R}}|C_S(t+\tau_n)-C_S(t)|\le12\pi\varphi^{-n} sup t ∈ R ∣ C S ( t + τ n ) − C S ( t ) ∣ ≤ 12 π φ − n
(iii) C S ( τ n ) = 1 + 4 cos ( 2 π φ − n ) + cos ( 4 π φ − n ) CS(\tau_n)=1+4\cos(2\pi\varphi^{-n})+\cos(4\pi\varphi^{-n}) C S ( τ n ) = 1 + 4 cos ( 2 π φ − n ) + cos ( 4 π φ − n ) C S ( τ n ) < 6 CS(\tau_n)<6 C S ( τ n ) < 6 n ≥ 1 n\ge1 n ≥ 1 0 < 6 − C S ( τ n ) ≤ 16 π 2 φ − 2 n 0<6-CS(\tau_n)\le16\pi^2\varphi^{-2n} 0 < 6 − C S ( τ n ) ≤ 16 π 2 φ − 2 n
Thus the maximum value 6, attained only at t = 0 t=0 t = 0 τ n → ∞ \tau_n\to\infty τ n → ∞
Proof. (i) Let ϵ n = Fib n + 1 − φ Fib n \epsilon_n=\operatorname{Fib}_{n+1}-\varphi\operatorname{Fib}_n ϵ n = Fib n + 1 − φ Fib n ϵ 0 = 1 \epsilon_0=1 ϵ 0 = 1 1 − φ = − φ − 1 1-\varphi=-\varphi^{-1} 1 − φ = − φ − 1 ϵ n + 1 = Fib n + 1 + Fib n − φ Fib n + 1 = − φ − 1 Fib n + 1 + Fib n = − φ − 1 ϵ n \epsilon_{n+1}=\operatorname{Fib}_{n+1}+\operatorname{Fib}_n-\varphi\operatorname{Fib}_{n+1}=-\varphi^{-1}\operatorname{Fib}_{n+1}+\operatorname{Fib}_n=-\varphi^{-1}\epsilon_n ϵ n + 1 = Fib n + 1 + Fib n − φ Fib n + 1 = − φ − 1 Fib n + 1 + Fib n = − φ − 1 ϵ n ϵ n = ( − φ − 1 ) n \epsilon_n=(-\varphi^{-1})^n ϵ n = ( − φ − 1 ) n ∥ φ Fib n ∥ ≤ ∣ φ Fib n − Fib n + 1 ∣ = φ − n \lVert\varphi\operatorname{Fib}_n\rVert\le|\varphi\operatorname{Fib}_n-\operatorname{Fib}_{n+1}|=\varphi^{-n} ∥ φ Fib n ∥ ≤ ∣ φ Fib n − Fib n + 1 ∣ = φ − n n ≥ 2 n\ge2 n ≥ 2 φ − n ≤ φ − 2 = 2 − φ < 1 2 \varphi^{-n}\le\varphi^{-2}=2-\varphi<\frac12 φ − n ≤ φ − 2 = 2 − φ < 2 1 Fib n + 1 \operatorname{Fib}_{n+1} Fib n + 1 φ Fib n \varphi\operatorname{Fib}_n φ Fib n n = 1 n=1 n = 1 ∥ φ ∥ = 2 − φ = φ − 2 \lVert\varphi\rVert=2-\varphi=\varphi^{-2} ∥ φ ∥ = 2 − φ = φ − 2 u τ n = Fib n ∈ Z u\tau_n=\operatorname{Fib}_n\in\mathbb{Z} u τ n = Fib n ∈ Z ∥ u τ n ∥ = 0 \lVert u\tau_n\rVert=0 ∥ u τ n ∥ = 0 ∥ φ u τ n ∥ ≤ φ − n \lVert\varphi u\tau_n\rVert\le\varphi^{-n} ∥ φ u τ n ∥ ≤ φ − n Theorem 4 (b). (iii) Since θ ↦ cos 2 π θ \theta\mapsto\cos2\pi\theta θ ↦ cos 2 π θ cos ( 2 π Fib n ) = 1 \cos(2\pi\operatorname{Fib}_n)=1 cos ( 2 π Fib n ) = 1 cos ( 2 π φ Fib n ) = cos ( 2 π ( Fib n + 1 − ϵ n ) ) = cos ( 2 π φ − n ) \cos(2\pi\varphi\operatorname{Fib}_n)=\cos(2\pi(\operatorname{Fib}_{n+1}-\epsilon_n))=\cos(2\pi\varphi^{-n}) cos ( 2 π φ Fib n ) = cos ( 2 π ( Fib n + 1 − ϵ n )) = cos ( 2 π φ − n ) cos ( 2 π φ 2 Fib n ) = cos ( 2 π ( φ Fib n + Fib n ) ) = cos ( 2 π φ Fib n ) \cos(2\pi\varphi^2\operatorname{Fib}_n)=\cos(2\pi(\varphi\operatorname{Fib}_n+\operatorname{Fib}_n))=\cos(2\pi\varphi\operatorname{Fib}_n) cos ( 2 π φ 2 Fib n ) = cos ( 2 π ( φ Fib n + Fib n )) = cos ( 2 π φ Fib n ) cos ( 2 π φ 3 Fib n ) = cos ( 2 π ( 2 φ Fib n + Fib n ) ) = cos ( 4 π ϵ n ) = cos ( 4 π φ − n ) \cos(2\pi\varphi^3\operatorname{Fib}_n)=\cos(2\pi(2\varphi\operatorname{Fib}_n+\operatorname{Fib}_n))=\cos(4\pi\epsilon_n)=\cos(4\pi\varphi^{-n}) cos ( 2 π φ 3 Fib n ) = cos ( 2 π ( 2 φ Fib n + Fib n )) = cos ( 4 π ϵ n ) = cos ( 4 π φ − n ) n ≥ 1 n\ge1 n ≥ 1 τ n ≠ 0 \tau_n\ne0 τ n = 0 C S ( τ n ) < 6 CS(\tau_n)<6 C S ( τ n ) < 6 Proposition 6.2 (ii). Finally, with ϑ = φ − n \vartheta=\varphi^{-n} ϑ = φ − n 1 − cos y ≤ y 2 / 2 1-\cos y\le y^2/2 1 − cos y ≤ y 2 /2 6 − C S ( τ n ) = 4 ( 1 − cos 2 π ϑ ) + ( 1 − cos 4 π ϑ ) ≤ 8 π 2 ϑ 2 + 8 π 2 ϑ 2 6-CS(\tau_n)=4(1-\cos2\pi\vartheta)+(1-\cos4\pi\vartheta)\le8\pi^2\vartheta^2+8\pi^2\vartheta^2 6 − C S ( τ n ) = 4 ( 1 − cos 2 π ϑ ) + ( 1 − cos 4 π ϑ ) ≤ 8 π 2 ϑ 2 + 8 π 2 ϑ 2
Lemma 6.6 (density). For u > 0 u>0 u > 0 { ( u t , φ u t ) m o d Z 2 : t ∈ R } \{(ut,\varphi ut)\bmod\mathbb{Z}^2:t\in\mathbb{R}\} {( u t , φ u t ) mod Z 2 : t ∈ R } T 2 \mathbb{T}^2 T 2
Proof. Let ( ξ 1 , ξ 2 ) ∈ R 2 (\xi_1,\xi_2)\in\mathbb{R}^2 ( ξ 1 , ξ 2 ) ∈ R 2 η > 0 \eta>0 η > 0 t = ( ξ 1 + k ) / u t=(\xi_1+k)/u t = ( ξ 1 + k ) / u k ∈ Z k\in\mathbb{Z} k ∈ Z ( ξ 1 , φ ξ 1 + k φ ) m o d Z 2 (\xi_1,\varphi\xi_1+k\varphi)\bmod\mathbb{Z}^2 ( ξ 1 , φ ξ 1 + k φ ) mod Z 2 { k φ m o d 1 : k ∈ Z } \{k\varphi\bmod1:k\in\mathbb{Z}\} { k φ mod 1 : k ∈ Z } T \mathbb{T} T N > 1 / η N>1/\eta N > 1/ η k φ k\varphi k φ 0 ≤ k ≤ N 0\le k\le N 0 ≤ k ≤ N N + 1 N+1 N + 1 [ 0 , 1 ) [0,1) [ 0 , 1 ) φ ∉ Q \varphi\notin\mathbb{Q} φ ∈ / Q k 1 < k 2 k_1<k_2 k 1 < k 2 1 / N 1/N 1/ N m = k 2 − k 1 ≥ 1 m=k_2-k_1\ge1 m = k 2 − k 1 ≥ 1 0 < δ : = ∥ m φ ∥ < η 0<\delta:=\lVert m\varphi\rVert<\eta 0 < δ := ∥ m φ ∥ < η j m φ m o d 1 jm\varphi\bmod1 j m φ mod 1 j ∈ Z j\in\mathbb{Z} j ∈ Z ± j δ m o d 1 \pm j\delta\bmod1 ± j δ mod 1 δ \delta δ j j j ∥ j m φ − ( ξ 2 − φ ξ 1 ) ∥ ≤ δ / 2 < η \lVert jm\varphi-(\xi_2-\varphi\xi_1)\rVert\le\delta/2<\eta ∥ j m φ − ( ξ 2 − φ ξ 1 )∥ ≤ δ /2 < η
This is the special case of Kronecker’s theorem [25 ] needed below.
Proposition 6.8 (sharp constant). For n ≥ 1 n\ge1 n ≥ 1 τ n = Fib n / u \tau_n=\operatorname{Fib}_n/u τ n = Fib n / u E n = sup t ∈ R ∣ C S ( t + τ n ) − C S ( t ) ∣ E_n=\sup_{t\in\mathbb{R}}|CS(t+\tau_n)-CS(t)| E n = sup t ∈ R ∣ C S ( t + τ n ) − C S ( t ) ∣ δ n = φ − n \delta_n=\varphi^{-n} δ n = φ − n − 1 ≤ γ ≤ 1 -1\le\gamma\le1 − 1 ≤ γ ≤ 1 μ ( γ ) = max X , Y ∈ R ∣ sin X + sin Y + γ sin ( X + Y ) ∣ \mu(\gamma)=\max_{X,Y\in\mathbb{R}}|\sin X+\sin Y+\gamma\sin(X+Y)| μ ( γ ) = max X , Y ∈ R ∣ sin X + sin Y + γ sin ( X + Y ) ∣
(i) μ ( − γ ) = μ ( γ ) \mu(-\gamma)=\mu(\gamma) μ ( − γ ) = μ ( γ ) 0 ≤ γ ≤ 1 0\le\gamma\le1 0 ≤ γ ≤ 1 [22 (19)–(21)] and Remark 6.9)
μ ( γ ) = max 0 ≤ s ≤ 1 2 1 − s 2 ( 1 + γ s ) ; \mu(\gamma)=\max_{0\le s\le1}2\sqrt{1-s^2}(1+\gamma s); μ ( γ ) = 0 ≤ s ≤ 1 max 2 1 − s 2 ( 1 + γ s ) ; for γ > 0 \gamma>0 γ > 0 s ∗ ( γ ) = ( 1 + 8 γ 2 − 1 ) / ( 4 γ ) s^*(\gamma)=(\sqrt{1+8\gamma^2}-1)/(4\gamma) s ∗ ( γ ) = ( 1 + 8 γ 2 − 1 ) / ( 4 γ ) μ ( 0 ) = 2 \mu(0)=2 μ ( 0 ) = 2 μ \mu μ [ 0 , 1 ] [0,1] [ 0 , 1 ] μ ( 1 ) = 3 3 2 \mu(1)=\frac{3\sqrt{3}}{2} μ ( 1 ) = 2 3 3 X = Y = π / 3 X=Y=\pi/3 X = Y = π /3
(ii) E n = 4 sin ( π δ n ) μ ( cos π δ n ) E_n=4\sin(\pi\delta_n)\mu(\cos\pi\delta_n) E n = 4 sin ( π δ n ) μ ( cos π δ n ) n ≥ 1 n\ge1 n ≥ 1 E n ≤ 6 3 π φ − n E_n\le6\sqrt{3}\pi\varphi^{-n} E n ≤ 6 3 π φ − n
(iii) For n ≥ 2 n\ge2 n ≥ 2
E n = 6 3 π φ − n − 2 3 π 3 φ − 3 n + R n , 3 6 π 5 φ − 5 n ≤ R n ≤ ( 3 8 + 3 3 10 ) π 5 φ − 5 n . (1) E_n=6\sqrt{3}\pi\varphi^{-n}-2\sqrt{3}\pi^3\varphi^{-3n}+R_n,\qquad\frac{\sqrt{3}}{6}\pi^5\varphi^{-5n}\le R_n\le\left(\frac{3}{8}+\frac{3\sqrt{3}}{10}\right)\pi^5\varphi^{-5n}. \tag*{(1)} E n = 6 3 π φ − n − 2 3 π 3 φ − 3 n + R n , 6 3 π 5 φ − 5 n ≤ R n ≤ ( 8 3 + 10 3 3 ) π 5 φ − 5 n . (1) (iv) R n = 16 3 45 π 5 φ − 5 n + O ( φ − 7 n ) R_n=\frac{16\sqrt{3}}{45}\pi^5\varphi^{-5n}+O(\varphi^{-7n}) R n = 45 16 3 π 5 φ − 5 n + O ( φ − 7 n )
In particular the leading constant 6 3 π 6\sqrt{3}\,\pi 6 3 π 2 3 π 3 2\sqrt{3}\,\pi^3 2 3 π 3 6 3 π = 4 π μ ( 1 ) 6\sqrt{3}\,\pi=4\pi\mu(1) 6 3 π = 4 π μ ( 1 ) sup T 2 ∣ ∂ θ 2 Γ ∣ \sup_{\mathbb{T}^2}|\partial_{\theta_2}\Gamma| sup T 2 ∣ ∂ θ 2 Γ∣ Γ \Gamma Γ
Proof. (i) The substitution ( X , Y ) ↦ ( X + π , Y + π ) (X,Y)\mapsto(X+\pi,Y+\pi) ( X , Y ) ↦ ( X + π , Y + π ) sin X \sin X sin X sin Y \sin Y sin Y sin ( X + Y ) \sin(X+Y) sin ( X + Y ) μ ( − γ ) = μ ( γ ) \mu(-\gamma)=\mu(\gamma) μ ( − γ ) = μ ( γ ) 0 ≤ γ ≤ 1 0\leq\gamma\leq1 0 ≤ γ ≤ 1 Σ = X + Y \Sigma=X+Y Σ = X + Y D = ( X − Y ) / 2 D=(X-Y)/2 D = ( X − Y ) /2 ( X , Y ) (X,Y) ( X , Y ) R 2 \mathbb{R}^2 R 2 ( Σ , D ) (\Sigma,D) ( Σ , D ) sin X + sin Y = 2 sin ( Σ / 2 ) cos D \sin X+\sin Y=2\sin(\Sigma/2)\cos D sin X + sin Y = 2 sin ( Σ/2 ) cos D sin Σ = 2 sin ( Σ / 2 ) cos ( Σ / 2 ) \sin\Sigma=2\sin(\Sigma/2)\cos(\Sigma/2) sin Σ = 2 sin ( Σ/2 ) cos ( Σ/2 )
sin X + sin Y + γ sin ( X + Y ) = 2 sin Σ 2 ( cos D + γ cos Σ 2 ) . \sin X+\sin Y+\gamma\sin(X+Y)=2\sin\frac{\Sigma}{2}\left(\cos D+\gamma\cos\frac{\Sigma}{2}\right). sin X + sin Y + γ sin ( X + Y ) = 2 sin 2 Σ ( cos D + γ cos 2 Σ ) . For fixed Σ \Sigma Σ D D D 2 ∣ sin ( Σ / 2 ) ∣ ( 1 + γ ∣ cos ( Σ / 2 ) ∣ ) 2|\sin(\Sigma/2)|(1+\gamma|\cos(\Sigma/2)|) 2∣ sin ( Σ/2 ) ∣ ( 1 + γ ∣ cos ( Σ/2 ) ∣ ) cos D = ± 1 \cos D=\pm1 cos D = ± 1 cos ( Σ / 2 ) \cos(\Sigma/2) cos ( Σ/2 ) s = ∣ cos ( Σ / 2 ) ∣ s=|\cos(\Sigma/2)| s = ∣ cos ( Σ/2 ) ∣ [ 0 , 1 ] [0,1] [ 0 , 1 ] ∣ sin ( Σ / 2 ) ∣ = 1 − s 2 |\sin(\Sigma/2)|=\sqrt{1-s^2} ∣ sin ( Σ/2 ) ∣ = 1 − s 2 γ ↦ 2 1 − s 2 ( 1 + γ s ) \gamma\mapsto2\sqrt{1-s^2}(1+\gamma s) γ ↦ 2 1 − s 2 ( 1 + γ s ) μ \mu μ γ > 0 \gamma>0 γ > 0 0 ≤ s < 1 0\leq s<1 0 ≤ s < 1
d d s [ 1 − s 2 ( 1 + γ s ) ] = γ − s − 2 γ s 2 1 − s 2 , \frac{d}{ds}\left[\sqrt{1-s^2}(1+\gamma s)\right]=\frac{\gamma-s-2\gamma s^2}{\sqrt{1-s^2}}, d s d [ 1 − s 2 ( 1 + γ s ) ] = 1 − s 2 γ − s − 2 γ s 2 , which is positive for s < s ∗ ( γ ) s<s^*(\gamma) s < s ∗ ( γ ) s > s ∗ ( γ ) s>s^*(\gamma) s > s ∗ ( γ ) s ∗ ( γ ) ∈ ( 0 , 1 ) s^*(\gamma)\in(0,1) s ∗ ( γ ) ∈ ( 0 , 1 ) 2 γ s 2 + s − γ = 0 2\gamma s^2+s-\gamma=0 2 γ s 2 + s − γ = 0 γ = 1 \gamma=1 γ = 1 s ∗ = 1 2 s^*=\frac12 s ∗ = 2 1 μ ( 1 ) = 2 ⋅ 3 2 ⋅ 3 2 = 3 3 2 \mu(1)=2\cdot\frac{\sqrt{3}}{2}\cdot\frac32=\frac{3\sqrt{3}}{2} μ ( 1 ) = 2 ⋅ 2 3 ⋅ 2 3 = 2 3 3 s = 1 2 s=\frac12 s = 2 1 D = 0 D=0 D = 0 Σ = 2 π / 3 \Sigma=2\pi/3 Σ = 2 π /3 X = Y = π / 3 X=Y=\pi/3 X = Y = π /3
(ii) Let ϵ n = Fib n + 1 − φ Fib n = ( − φ − 1 ) n \epsilon_n=\operatorname{Fib}_{n+1}-\varphi\operatorname{Fib}_n=(-\varphi^{-1})^n ϵ n = Fib n + 1 − φ Fib n = ( − φ − 1 ) n Proposition 6.5 (i)) and δ = − ϵ n \delta=-\epsilon_n δ = − ϵ n ∣ δ ∣ = δ n |\delta|=\delta_n ∣ δ ∣ = δ n Γ \Gamma Γ ( θ 1 , θ 2 ) = ( u t , φ u t ) (\theta_1,\theta_2)=(ut,\varphi ut) ( θ 1 , θ 2 ) = ( u t , φ u t ) u ( t + τ n ) = θ 1 + Fib n u(t+\tau_n)=\theta_1+\operatorname{Fib}_n u ( t + τ n ) = θ 1 + Fib n φ u ( t + τ n ) = θ 2 + Fib n + 1 + δ \varphi u(t+\tau_n)=\theta_2+\operatorname{Fib}_{n+1}+\delta φ u ( t + τ n ) = θ 2 + Fib n + 1 + δ 1 1 1 Γ \Gamma Γ C S ( t + τ n ) − C S ( t ) = Γ ( θ 1 , θ 2 + δ ) − Γ ( θ 1 , θ 2 ) CS(t+\tau_n)-CS(t)=\Gamma(\theta_1,\theta_2+\delta)-\Gamma(\theta_1,\theta_2) C S ( t + τ n ) − C S ( t ) = Γ ( θ 1 , θ 2 + δ ) − Γ ( θ 1 , θ 2 ) θ 2 \theta_2 θ 2 Γ \Gamma Γ 0 , 1 , 1 , 2 0,1,1,2 0 , 1 , 1 , 2 cos ( α + 2 k π δ ) − cos α = − 2 sin ( k π δ ) sin ( α + k π δ ) \cos(\alpha+2k\pi\delta)-\cos\alpha=-2\sin(k\pi\delta)\sin(\alpha+k\pi\delta) cos ( α + 2 k π δ ) − cos α = − 2 sin ( k π δ ) sin ( α + k π δ ) sin 2 π δ = 2 sin π δ cos π δ \sin2\pi\delta=2\sin\pi\delta\cos\pi\delta sin 2 π δ = 2 sin π δ cos π δ X = 2 π θ 2 + π δ X=2\pi\theta_2+\pi\delta X = 2 π θ 2 + π δ Y = 2 π ( θ 1 + θ 2 ) + π δ Y=2\pi(\theta_1+\theta_2)+\pi\delta Y = 2 π ( θ 1 + θ 2 ) + π δ 2 π ( θ 1 + 2 θ 2 ) + 2 π δ = X + Y 2\pi(\theta_1+2\theta_2)+2\pi\delta=X+Y 2 π ( θ 1 + 2 θ 2 ) + 2 π δ = X + Y
Γ ( θ 1 , θ 2 + δ ) − Γ ( θ 1 , θ 2 ) = − 4 sin ( π δ ) ( sin X + sin Y + cos ( π δ ) sin ( X + Y ) ) . \Gamma(\theta_1,\theta_2+\delta)-\Gamma(\theta_1,\theta_2)=-4\sin(\pi\delta)\left(\sin X+\sin Y+\cos(\pi\delta)\sin(X+Y)\right). Γ ( θ 1 , θ 2 + δ ) − Γ ( θ 1 , θ 2 ) = − 4 sin ( π δ ) ( sin X + sin Y + cos ( π δ ) sin ( X + Y ) ) . The map ( θ 1 , θ 2 ) ↦ ( X , Y ) (\theta_1,\theta_2)\mapsto(X,Y) ( θ 1 , θ 2 ) ↦ ( X , Y ) 2 π ( 0 1 1 1 ) 2\pi\begin{pmatrix}0&1\\1&1\end{pmatrix} 2 π ( 0 1 1 1 ) ( 0 1 1 1 ) \begin{pmatrix}0&1\\1&1\end{pmatrix} ( 0 1 1 1 ) − 1 -1 − 1 T 2 → ( R / 2 π Z ) 2 \mathbb{T}^2\to(\mathbb{R}/2\pi\mathbb{Z})^2 T 2 → ( R /2 π Z ) 2 Lemma 6.6 the image of t ↦ ( θ 1 , θ 2 ) t\mapsto(\theta_1,\theta_2) t ↦ ( θ 1 , θ 2 ) t ↦ ( X , Y ) t\mapsto(X,Y) t ↦ ( X , Y ) ( X , Y ) ↦ ∣ sin X + sin Y + γ sin ( X + Y ) ∣ (X,Y)\mapsto|\sin X+\sin Y+\gamma\sin(X+Y)| ( X , Y ) ↦ ∣ sin X + sin Y + γ sin ( X + Y ) ∣ ( R / 2 π Z ) 2 (\mathbb{R}/2\pi\mathbb{Z})^2 ( R /2 π Z ) 2 μ ( γ ) \mu(\gamma) μ ( γ ) 0 < δ n < 1 0<\delta_n<1 0 < δ n < 1 ∣ sin π δ ∣ = sin π δ n |\sin\pi\delta|=\sin\pi\delta_n ∣ sin π δ ∣ = sin π δ n cos π δ = cos π δ n \cos\pi\delta=\cos\pi\delta_n cos π δ = cos π δ n μ ( cos π δ n ) ≤ μ ( 1 ) = 3 3 2 \mu(\cos\pi\delta_n)\leq\mu(1)=\frac{3\sqrt{3}}{2} μ ( cos π δ n ) ≤ μ ( 1 ) = 2 3 3 sin y ≤ y \sin y\leq y sin y ≤ y E n ≤ 6 3 π δ n E_n\leq6\sqrt{3}\,\pi\delta_n E n ≤ 6 3 π δ n
(iii) Let n ≥ 2 n\geq2 n ≥ 2 y = π δ n ≤ π φ − 2 < 1.2 y=\pi\delta_n\leq\pi\varphi^{-2}<1.2 y = π δ n ≤ π φ − 2 < 1.2 γ = cos y ∈ ( 0 , 1 ) \gamma=\cos y\in(0,1) γ = cos y ∈ ( 0 , 1 )
3 3 2 − 3 2 ( 1 − γ ) ≤ μ ( γ ) ≤ 3 3 2 − 3 2 ( 1 − γ ) + 3 8 ( 1 − γ ) 2 . (2) \frac{3\sqrt{3}}{2}-\frac{\sqrt{3}}{2}(1-\gamma)\leq\mu(\gamma)\leq\frac{3\sqrt{3}}{2}-\frac{\sqrt{3}}{2}(1-\gamma)+\frac{3}{8}(1-\gamma)^2. \tag*{(2)} 2 3 3 − 2 3 ( 1 − γ ) ≤ μ ( γ ) ≤ 2 3 3 − 2 3 ( 1 − γ ) + 8 3 ( 1 − γ ) 2 . (2) The lower bound is the value at s = 1 2 s=\frac12 s = 2 1 2 1 − s 2 ( 1 + γ s ) = f ( s ) − ( 1 − γ ) g ( s ) 2\sqrt{1-s^2}(1+\gamma s)=f(s)-(1-\gamma)g(s) 2 1 − s 2 ( 1 + γ s ) = f ( s ) − ( 1 − γ ) g ( s ) f ( s ) = 2 1 − s 2 ( 1 + s ) f(s)=2\sqrt{1-s^2}(1+s) f ( s ) = 2 1 − s 2 ( 1 + s ) g ( s ) = 2 s 1 − s 2 g(s)=2s\sqrt{1-s^2} g ( s ) = 2 s 1 − s 2 [ 0 , 1 ] [0,1] [ 0 , 1 ] 3 3 2 − 2 ( s − 1 2 ) 2 ≥ 3 3 2 − 1 2 > 0 \frac{3\sqrt{3}}{2}-2(s-\frac12)^2\geq\frac{3\sqrt{3}}{2}-\frac12>0 2 3 3 − 2 ( s − 2 1 ) 2 ≥ 2 3 3 − 2 1 > 0
( 3 3 2 − 2 ( s − 1 2 ) 2 ) 2 − f ( s ) 2 = ( s − 1 2 ) 2 ( 8 s 2 + 8 s + 12 − 6 3 ) ≥ 0 , \left(\frac{3\sqrt{3}}{2}-2\left(s-\frac12\right)^2\right)^2-f(s)^2=\left(s-\frac12\right)^2(8s^2+8s+12-6\sqrt{3})\geq0, ( 2 3 3 − 2 ( s − 2 1 ) 2 ) 2 − f ( s ) 2 = ( s − 2 1 ) 2 ( 8 s 2 + 8 s + 12 − 6 3 ) ≥ 0 , so f ( s ) ≤ 3 3 2 − 2 ( s − 1 2 ) 2 f(s)\leq\frac{3\sqrt{3}}{2}-2(s-\frac12)^2 f ( s ) ≤ 2 3 3 − 2 ( s − 2 1 ) 2 g ′ ′ ( s ) = 2 s ( 2 s 2 − 3 ) ( 1 − s 2 ) − 3 / 2 ≤ 0 g''(s)=2s(2s^2-3)(1-s^2)^{-3/2}\leq0 g ′′ ( s ) = 2 s ( 2 s 2 − 3 ) ( 1 − s 2 ) − 3/2 ≤ 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] g g g [ 0 , 1 ] [0,1] [ 0 , 1 ] g ( 0 ) = g ( 1 ) = 0 g(0)=g(1)=0 g ( 0 ) = g ( 1 ) = 0 g ( 1 2 ) = 3 2 g(\frac12)=\frac{\sqrt{3}}{2} g ( 2 1 ) = 2 3 g ( s ) ≥ 3 2 − 3 ∣ s − 1 2 ∣ g(s)\geq\frac{\sqrt{3}}{2}-\sqrt{3}|s-\frac12| g ( s ) ≥ 2 3 − 3 ∣ s − 2 1 ∣ d = ∣ s − 1 2 ∣ d=|s-\frac12| d = ∣ s − 2 1 ∣
2 1 − s 2 ( 1 + γ s ) ≤ 3 3 2 − 3 2 ( 1 − γ ) + ( 3 ( 1 − γ ) d − 2 d 2 ) ≤ 3 3 2 − 3 2 ( 1 − γ ) + 3 8 ( 1 − γ ) 2 , 2\sqrt{1-s^2}(1+\gamma s)\leq\frac{3\sqrt{3}}{2}-\frac{\sqrt{3}}{2}(1-\gamma)+(\sqrt{3}(1-\gamma)d-2d^2)\leq\frac{3\sqrt{3}}{2}-\frac{\sqrt{3}}{2}(1-\gamma)+\frac{3}{8}(1-\gamma)^2, 2 1 − s 2 ( 1 + γ s ) ≤ 2 3 3 − 2 3 ( 1 − γ ) + ( 3 ( 1 − γ ) d − 2 d 2 ) ≤ 2 3 3 − 2 3 ( 1 − γ ) + 8 3 ( 1 − γ ) 2 , which is (1). Now use y − y 3 6 ≤ sin y ≤ y − y 3 6 + y 5 120 y-\frac{y^3}{6}\le\sin y\le y-\frac{y^3}{6}+\frac{y^5}{120} y − 6 y 3 ≤ sin y ≤ y − 6 y 3 + 120 y 5 y 2 2 − y 4 24 ≤ 1 − cos y ≤ y 2 2 \frac{y^2}{2}-\frac{y^4}{24}\le1-\cos y\le\frac{y^2}{2} 2 y 2 − 24 y 4 ≤ 1 − cos y ≤ 2 y 2 y ≥ 0 y\ge0 y ≥ 0 0 < y < 1.2 0<y<1.2 0 < y < 1.2
E n ≥ 4 ( y − y 3 6 ) ( 3 3 2 − 3 4 y 2 ) = 6 3 y − 2 3 y 3 + 3 6 y 5 , E_n\ge4\left(y-\frac{y^3}{6}\right)\left(\frac{3\sqrt{3}}{2}-\frac{\sqrt{3}}{4}y^2\right)=6\sqrt{3}y-2\sqrt{3}y^3+\frac{\sqrt{3}}{6}y^5, E n ≥ 4 ( y − 6 y 3 ) ( 2 3 3 − 4 3 y 2 ) = 6 3 y − 2 3 y 3 + 6 3 y 5 , E n ≤ 4 ( y − y 3 6 + y 5 120 ) ( 3 3 2 − 3 4 y 2 + ( 3 48 + 3 32 ) y 4 ) = 6 3 y − 2 3 y 3 + q ( y ) y 5 , E_n\le4\left(y-\frac{y^3}{6}+\frac{y^5}{120}\right)\left(\frac{3\sqrt{3}}{2}-\frac{\sqrt{3}}{4}y^2+\left(\frac{\sqrt{3}}{48}+\frac{3}{32}\right)y^4\right)=6\sqrt{3}y-2\sqrt{3}y^3+q(y)y^5, E n ≤ 4 ( y − 6 y 3 + 120 y 5 ) ( 2 3 3 − 4 3 y 2 + ( 48 3 + 32 3 ) y 4 ) = 6 3 y − 2 3 y 3 + q ( y ) y 5 , where q ( y ) = 3 8 + 3 3 10 − ( 1 16 + 3 45 ) y 2 + ( 1 320 + 3 1440 ) y 4 ≤ 3 10 + 3 3 10 q(y)=\frac{3}{8}+\frac{3\sqrt{3}}{10}-\left(\frac{1}{16}+\frac{\sqrt{3}}{45}\right)y^2+\left(\frac{1}{320}+\frac{\sqrt{3}}{1440}\right)y^4\le\frac{3}{10}+\frac{3\sqrt{3}}{10} q ( y ) = 8 3 + 10 3 3 − ( 16 1 + 45 3 ) y 2 + ( 320 1 + 1440 3 ) y 4 ≤ 10 3 + 10 3 3 0 < y ≤ 4 0<y\le4 0 < y ≤ 4 y = π φ − n y=\pi\varphi^{-n} y = π φ − n ∂ θ 2 Γ = − 4 π ( sin 2 π θ 2 + sin 2 π ( θ 1 + θ 2 ) + sin 2 π ( θ 1 + 2 θ 2 ) ) \partial_{\theta_2}\Gamma=-4\pi(\sin2\pi\theta_2+\sin2\pi(\theta_1+\theta_2)+\sin2\pi(\theta_1+2\theta_2)) ∂ θ 2 Γ = − 4 π ( sin 2 π θ 2 + sin 2 π ( θ 1 + θ 2 ) + sin 2 π ( θ 1 + 2 θ 2 )) 4 π μ ( 1 ) = 6 3 π 4\pi\mu(1)=6\sqrt{3}\pi 4 π μ ( 1 ) = 6 3 π
(iv) Near γ = 1 \gamma=1 γ = 1 s ∗ ( γ ) s^*(\gamma) s ∗ ( γ ) 2 γ s 2 + s − γ = 0 2\gamma s^2+s-\gamma=0 2 γ s 2 + s − γ = 0 μ ( γ ) = Ψ ( s ∗ ( γ ) , γ ) \mu(\gamma)=\Psi(s^*(\gamma),\gamma) μ ( γ ) = Ψ ( s ∗ ( γ ) , γ ) Ψ ( s , γ ) = 2 1 − s 2 ( 1 + γ s ) \Psi(s,\gamma)=2\sqrt{1-s^2}(1+\gamma s) Ψ ( s , γ ) = 2 1 − s 2 ( 1 + γ s ) ∂ s Ψ ( s ∗ ( γ ) , γ ) = 0 \partial_s\Psi(s^*(\gamma),\gamma)=0 ∂ s Ψ ( s ∗ ( γ ) , γ ) = 0 μ ′ = 2 s ∗ 1 − s ∗ 2 \mu'=2s^*\sqrt{1-s^{*2}} μ ′ = 2 s ∗ 1 − s ∗ 2 μ ′ ′ = 2 s ∗ ′ ( 1 − 2 s ∗ 2 ) / 1 − s ∗ 2 \mu''=2s^{*\prime}(1-2s^{*2})/\sqrt{1-s^{*2}} μ ′′ = 2 s ∗ ′ ( 1 − 2 s ∗ 2 ) / 1 − s ∗ 2 γ = 1 \gamma=1 γ = 1 s ∗ = 1 2 s^*=\frac{1}{2} s ∗ = 2 1 2 γ s ∗ 2 + s ∗ − γ = 0 2\gamma s^{*2}+s^*-\gamma=0 2 γ s ∗ 2 + s ∗ − γ = 0 s ∗ ′ ( 1 ) = ( 1 − 2 s ∗ 2 ) / ( 4 γ s ∗ + 1 ) = 1 6 s^{*\prime}(1)=(1-2s^{*2})/(4\gamma s^*+1)=\frac{1}{6} s ∗ ′ ( 1 ) = ( 1 − 2 s ∗ 2 ) / ( 4 γ s ∗ + 1 ) = 6 1 μ ′ ( 1 ) = 3 6 \mu'(1)=\frac{\sqrt{3}}{6} μ ′ ( 1 ) = 6 3 μ ′ ′ ( 1 ) = 3 9 \mu''(1)=\frac{\sqrt{3}}{9} μ ′′ ( 1 ) = 9 3 1 − cos y = y 2 2 − y 4 24 + O ( y 6 ) 1-\cos y=\frac{y^2}{2}-\frac{y^4}{24}+O(y^6) 1 − cos y = 2 y 2 − 24 y 4 + O ( y 6 )
μ ( cos y ) = 3 3 2 − 3 4 y 2 + 5 3 144 y 4 + O ( y 6 ) , \mu(\cos y)=\frac{3\sqrt{3}}{2}-\frac{\sqrt{3}}{4}y^2+\frac{5\sqrt{3}}{144}y^4+O(y^6), μ ( cos y ) = 2 3 3 − 4 3 y 2 + 144 5 3 y 4 + O ( y 6 ) , and multiplying by 4 sin y = 4 y − 2 3 y 3 + 1 30 y 5 + O ( y 7 ) 4\sin y=4y-\frac{2}{3}y^3+\frac{1}{30}y^5+O(y^7) 4 sin y = 4 y − 3 2 y 3 + 30 1 y 5 + O ( y 7 ) E n = 6 3 y − 2 3 y 3 + 16 3 45 y 5 + O ( y 7 ) E_n=6\sqrt{3}y-2\sqrt{3}y^3+\frac{16\sqrt{3}}{45}y^5+O(y^7) E n = 6 3 y − 2 3 y 3 + 45 16 3 y 5 + O ( y 7 ) y = π φ − n y=\pi\varphi^{-n} y = π φ − n
Remark 6.9. Writing s = cos θ s=\cos\theta s = cos θ 2 1 − s 2 ( 1 + γ s ) = 2 ( sin θ + γ 2 sin 2 θ ) 2\sqrt{1-s^2}(1+\gamma s)=2(\sin\theta+\frac{\gamma}{2}\sin2\theta) 2 1 − s 2 ( 1 + γ s ) = 2 ( sin θ + 2 γ sin 2 θ ) μ ( γ ) = 2 M ( γ / 2 ) \mu(\gamma)=2M(\gamma/2) μ ( γ ) = 2 M ( γ /2 ) M ( g ) = max θ ( sin θ + g sin 2 θ ) M(g)=\max_\theta(\sin\theta+g\sin2\theta) M ( g ) = max θ ( sin θ + g sin 2 θ ) [22 (19)–(21)] (their γ c + ( g ) \gamma_c+(g) γ c + ( g ) cos θ 0 = ( 1 + 32 g 2 − 1 ) / ( 8 g ) \cos\theta_0=(\sqrt{1+32g^2}-1)/(8g) cos θ 0 = ( 1 + 32 g 2 − 1 ) / ( 8 g ) s ∗ ( γ ) s^*(\gamma) s ∗ ( γ ) g = γ / 2 g=\gamma/2 g = γ /2 0 < γ ≤ 1 0<\gamma\le1 0 < γ ≤ 1 r = 1 + 8 γ 2 r=\sqrt{1+8\gamma^2} r = 1 + 8 γ 2 μ ( γ ) = ( r + 3 ) 3 / 2 ( r − 1 ) 1 / 2 / ( 8 γ ) \mu(\gamma)=(r+3)^{3/2}(r-1)^{1/2}/(8\gamma) μ ( γ ) = ( r + 3 ) 3/2 ( r − 1 ) 1/2 / ( 8 γ ) μ ( 1 ) = 432 / 8 = 3 3 / 2 \mu(1)=\sqrt{432}/8=3\sqrt{3}/2 μ ( 1 ) = 432 /8 = 3 3 /2
Remark 6.10. The upper bound E n ≤ 6 3 π φ − n E_n\le6\sqrt{3}\pi\varphi^{-n} E n ≤ 6 3 π φ − n ∣ δ ∣ sup ∣ ∂ θ 2 Γ ∣ |\delta|\sup|\partial_{\theta_2}\Gamma| ∣ δ ∣ sup ∣ ∂ θ 2 Γ∣ 12 π φ − n 12\pi\varphi^{-n} 12 π φ − n Proposition 6.5 (ii). By (iv), R n φ 5 n / π 5 → 16 3 / 45 = 0.61584 … R_n\varphi^{5n}/\pi^5\to16\sqrt{3}/45=0.61584\ldots R n φ 5 n / π 5 → 16 3 /45 = 0.61584 … § 7 ), inside the interval [ 0.289 , 0.895 ] [0.289,0.895] [ 0.289 , 0.895 ]
For ε > 0 \varepsilon>0 ε > 0 L ( ε ) L(\varepsilon) L ( ε ) L > 0 L>0 L > 0 L L L E ( C S ; ε ) E(CS;\varepsilon) E ( C S ; ε ) Definition 2.9 ).
Proposition 6.11 (order of L ( ε ) L(\varepsilon) L ( ε ) For 0 < ε ≤ 1 2 0<\varepsilon\le\frac{1}{2} 0 < ε ≤ 2 1
2 3 φ 2 u ⋅ 1 ε ≤ L ( ε ) ≤ 1 u ( 6 π φ 3 ε + 1 ) . \frac{\sqrt{2}}{3\varphi^2u}\cdot\frac{1}{\varepsilon}\le L(\varepsilon)\le\frac{1}{u}\left(\frac{6\pi\varphi^3}{\varepsilon}+1\right). 3 φ 2 u 2 ⋅ ε 1 ≤ L ( ε ) ≤ u 1 ( ε 6 π φ 3 + 1 ) . Proof. Upper bound. Let ε ≤ 1 \varepsilon\le1 ε ≤ 1 η = ε / ( 12 π ) \eta=\varepsilon/(12\pi) η = ε / ( 12 π ) m m m φ − m + 1 < 2 η \varphi^{-m+1}<2\eta φ − m + 1 < 2 η m ≥ 2 m\ge2 m ≥ 2 φ − m + 2 ≥ 2 η \varphi^{-m+2}\ge2\eta φ − m + 2 ≥ 2 η φ m ≤ φ 2 / ( 2 η ) \varphi^m\le\varphi^2/(2\eta) φ m ≤ φ 2 / ( 2 η ) Λ = { ( k , k φ − j ) : k , j ∈ Z } ⊂ R 2 \Lambda=\{(k,k\varphi-j):k,j\in\mathbb{Z}\}\subset\mathbb{R}^2 Λ = {( k , k φ − j ) : k , j ∈ Z } ⊂ R 2 Z 2 \mathbb{Z}^2 Z 2 ( k , j ) ↦ ( k , k φ − j ) (k,j)\mapsto(k,k\varphi-j) ( k , j ) ↦ ( k , k φ − j ) ( Fib m , Fib m + 1 ) (\operatorname{Fib}_m,\operatorname{Fib}_{m+1}) ( Fib m , Fib m + 1 ) ( Fib m + 1 , Fib m + 2 ) (\operatorname{Fib}_{m+1},\operatorname{Fib}_{m+2}) ( Fib m + 1 , Fib m + 2 ) Z 2 \mathbb{Z}^2 Z 2 v 1 = ( Fib m , − ϵ m ) v_1=(\operatorname{Fib}_m,-\epsilon_m) v 1 = ( Fib m , − ϵ m ) v 2 = ( Fib m + 1 , − ϵ m + 1 ) v_2=(\operatorname{Fib}_{m+1},-\epsilon_{m+1}) v 2 = ( Fib m + 1 , − ϵ m + 1 ) ϵ m = Fib m + 1 − φ Fib m = ( − φ ) − m \epsilon_m=\operatorname{Fib}_{m+1}-\varphi\operatorname{Fib}_m=(-\varphi)^{-m} ϵ m = Fib m + 1 − φ Fib m = ( − φ ) − m Λ \Lambda Λ P = { s v 1 + t v 2 : 0 ≤ s , t < 1 } P=\{sv_1+tv_2:0\le s,t<1\} P = { s v 1 + t v 2 : 0 ≤ s , t < 1 } Λ \Lambda Λ P P P [ 0 , Fib m + 2 ) [0,\operatorname{Fib}_{m+2}) [ 0 , Fib m + 2 ) ϵ m \epsilon_m ϵ m ϵ m + 1 \epsilon_{m+1} ϵ m + 1 ∣ ϵ m ∣ + ∣ ϵ m + 1 ∣ = φ − m + φ − m − 1 = φ − m + 1 < 2 η |\epsilon_m|+|\epsilon_{m+1}|=\varphi^{-m}+\varphi^{-m-1}=\varphi^{-m+1}<2\eta ∣ ϵ m ∣ + ∣ ϵ m + 1 ∣ = φ − m + φ − m − 1 = φ − m + 1 < 2 η k 0 k_0 k 0 [ k 0 , k 0 + Fib m + 2 ) × ( − η , η ) [k_0,k_0+\operatorname{Fib}_{m+2})\times(-\eta,\eta) [ k 0 , k 0 + Fib m + 2 ) × ( − η , η ) P P P Λ \Lambda Λ k ∈ [ k 0 , k 0 + Fib m + 2 ) k\in[k_0,k_0+\operatorname{Fib}_{m+2}) k ∈ [ k 0 , k 0 + Fib m + 2 ) ∥ k φ ∥ < η \|k\varphi\|<\eta ∥ k φ ∥ < η k k k τ = k / u \tau=k/u τ = k / u ∥ u τ ∥ = 0 \|u\tau\|=0 ∥ uτ ∥ = 0 ∥ φ u τ ∥ < η \|\varphi u\tau\|<\eta ∥ φ uτ ∥ < η sup t ∣ C S ( t + τ ) − C S ( t ) ∣ ≤ 12 π η = ε \sup_t|CS(t+\tau)-CS(t)|\le12\pi\eta=\varepsilon sup t ∣ C S ( t + τ ) − C S ( t ) ∣ ≤ 12 π η = ε Theorem 4 (b), i.e. τ ∈ E ( C S ; ε ) \tau\in E(CS;\varepsilon) τ ∈ E ( C S ; ε ) ( Fib m + 2 + 1 ) / u (\operatorname{Fib}_{m+2}+1)/u ( Fib m + 2 + 1 ) / u Fib m + 2 \operatorname{Fib}_{m+2} Fib m + 2 k / u k/u k / u Fib m + 2 ≤ φ m + 1 \operatorname{Fib}_{m+2}\le\varphi^{m+1} Fib m + 2 ≤ φ m + 1 F i b 1 ≤ φ 0 Fib_1 \le\varphi^0 F i b 1 ≤ φ 0 F i b 2 ≤ φ Fib_2 \le\varphi F i b 2 ≤ φ φ k + 1 = φ k + φ k − 1 \varphi^{k+1} = \varphi^k + \varphi^{k-1} φ k + 1 = φ k + φ k − 1 φ m + 1 ≤ φ 3 / ( 2 η ) = 6 π φ 3 / ε \varphi^{m+1} \le\varphi^3/(2\eta) = 6\pi\varphi^3/\varepsilon φ m + 1 ≤ φ 3 / ( 2 η ) = 6 π φ 3 / ε
Lower bound. For ω ∈ R 2 \omega\in\mathbb{R}^2 ω ∈ R 2 F ( ω ) = sup θ ∈ R 2 ∣ Γ ( θ + ω ) − Γ ( θ ) ∣ \mathcal{F}(\omega) = \sup_{\theta\in\mathbb{R}^2} |\Gamma(\theta+\omega)-\Gamma(\theta)| F ( ω ) = sup θ ∈ R 2 ∣Γ ( θ + ω ) − Γ ( θ ) ∣ Lemma 6.6 (and continuity of θ ↦ ∣ Γ ( θ + ω ) − Γ ( θ ) ∣ \theta\mapsto|\Gamma(\theta+\omega)-\Gamma(\theta)| θ ↦ ∣Γ ( θ + ω ) − Γ ( θ ) ∣ sup t ∣ C S ( t + τ ) − C S ( t ) ∣ = F ( u τ , φ u τ ) \sup_t |C_S(t+\tau)-C_S(t)| = \mathcal{F}(u\tau,\varphi u\tau) sup t ∣ C S ( t + τ ) − C S ( t ) ∣ = F ( uτ , φ uτ ) Γ ( θ ) = ∑ κ ∈ K 0 r κ cos ( 2 π κ ⋅ θ ) \Gamma(\theta) = \sum_{\kappa\in\mathcal{K}_0} r_\kappa\cos(2\pi\kappa\cdot\theta) Γ ( θ ) = ∑ κ ∈ K 0 r κ cos ( 2 π κ ⋅ θ ) K 0 = { ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) , ( 1 , 2 ) } \mathcal{K}_0 = \{(1,0),(0,1),(1,1),(1,2)\} K 0 = {( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) , ( 1 , 2 )} ( r κ ) = ( 1 , 2 , 2 , 1 ) (r_\kappa) = (1,2,2,1) ( r κ ) = ( 1 , 2 , 2 , 1 )
Γ ( θ + ω ) − Γ ( θ ) = − 2 ∑ κ ∈ K 0 r κ sin ( π κ ⋅ ω ) sin ( 2 π κ ⋅ θ + π κ ⋅ ω ) . \Gamma(\theta+\omega)-\Gamma(\theta) = -2 \sum_{\kappa\in\mathcal{K}_0} r_\kappa\sin(\pi\kappa\cdot\omega)\sin(2\pi\kappa\cdot\theta+\pi\kappa\cdot\omega). Γ ( θ + ω ) − Γ ( θ ) = − 2 κ ∈ K 0 ∑ r κ sin ( π κ ⋅ ω ) sin ( 2 π κ ⋅ θ + π κ ⋅ ω ) . No two elements of K 0 ∪ ( − K 0 ) \mathcal{K}_0\cup(-\mathcal{K}_0) K 0 ∪ ( − K 0 ) θ \theta θ L 2 ( T 2 ) L^2(\mathbb{T}^2) L 2 ( T 2 ) 2 − 1 / 2 2^{-1/2} 2 − 1/2 sup ∣ ψ ∣ ≥ ∥ ψ ∥ L 2 ( T 2 ) \sup|\psi| \ge\|\psi\|_{L^2(\mathbb{T}^2)} sup ∣ ψ ∣ ≥ ∥ ψ ∥ L 2 ( T 2 ) ∣ sin π x ∣ ≥ 2 ∥ x ∥ |\sin\pi x| \ge2\|x\| ∣ sin π x ∣ ≥ 2∥ x ∥ [ 0 , π / 2 ] [0,\pi/2] [ 0 , π /2 ]
F ( ω ) 2 ≥ 2 ∑ κ ∈ K 0 r κ 2 sin 2 ( π κ ⋅ ω ) ≥ 2 sin 2 ( π ω 1 ) + 8 sin 2 ( π ω 2 ) ≥ 8 ∥ ω 1 ∥ 2 + 32 ∥ ω 2 ∥ 2 ≥ 8 ∥ ω ∥ 2 , \mathcal{F}(\omega)^2 \ge2\sum_{\kappa\in\mathcal{K}_0} r_\kappa^2\sin^2(\pi\kappa\cdot\omega) \ge2\sin^2(\pi\omega_1)+8\sin^2(\pi\omega_2) \ge8\|\omega_1\|^2+32\|\omega_2\|^2 \ge8\|\omega\|^2, F ( ω ) 2 ≥ 2 κ ∈ K 0 ∑ r κ 2 sin 2 ( π κ ⋅ ω ) ≥ 2 sin 2 ( π ω 1 ) + 8 sin 2 ( π ω 2 ) ≥ 8∥ ω 1 ∥ 2 + 32∥ ω 2 ∥ 2 ≥ 8∥ ω ∥ 2 , where ∥ ω ∥ = max ( ∥ ω 1 ∥ , ∥ ω 2 ∥ ) \|\omega\|=\max(\|\omega_1\|,\|\omega_2\|) ∥ ω ∥ = max ( ∥ ω 1 ∥ , ∥ ω 2 ∥ ) F ( ω ) ≥ c ∥ ω ∥ \mathcal{F}(\omega)\ge c\|\omega\| F ( ω ) ≥ c ∥ ω ∥ c = 2 2 c=2\sqrt{2} c = 2 2
Let τ ∈ E ( C S ; ε ) \tau\in E(C_S;\varepsilon) τ ∈ E ( C S ; ε ) τ > 0 \tau>0 τ > 0 ∥ u τ ∥ , ∥ φ u τ ∥ ≤ ε / c \|u\tau\|,\|\varphi u\tau\|\le\varepsilon/c ∥ uτ ∥ , ∥ φ uτ ∥ ≤ ε / c u τ = k + s u\tau=k+s uτ = k + s k ∈ Z k\in\mathbb{Z} k ∈ Z ∣ s ∣ ≤ ε / c ≤ 1 / ( 4 2 ) < 1 |s|\le\varepsilon/c\le1/(4\sqrt{2})<1 ∣ s ∣ ≤ ε / c ≤ 1/ ( 4 2 ) < 1 u τ > 0 u\tau>0 uτ > 0 k ≥ 0 k\ge0 k ≥ 0 k = 0 k=0 k = 0 u τ ≤ ε / c u\tau\le\varepsilon/c uτ ≤ ε / c k ≥ 1 k\ge1 k ≥ 1 ∥ k φ ∥ ≤ ∥ φ u τ ∥ + φ ∣ s ∣ ≤ φ 2 ε / c \|k\varphi\|\le\|\varphi u\tau\|+\varphi|s|\le\varphi^2\varepsilon/c ∥ k φ ∥ ≤ ∥ φ uτ ∥ + φ ∣ s ∣ ≤ φ 2 ε / c ∥ k φ ∥ > 1 / ( 3 k ) \|k\varphi\|>1/(3k) ∥ k φ ∥ > 1/ ( 3 k ) j j j k φ k\varphi k φ ( j − k φ ) ( j − k φ ‾ ) = j 2 − j k − k 2 (j-k\varphi)(j-k\overline{\varphi})=j^2-jk-k^2 ( j − k φ ) ( j − k φ ) = j 2 − j k − k 2 ∣ j − k φ ‾ ∣ = ∣ j − k φ + k 5 ∣ ≤ 1 2 + 5 k < 3 k |j-k\overline{\varphi}|=|j-k\varphi+k\sqrt{5}|\le\frac{1}{2}+\sqrt{5}k<3k ∣ j − k φ ∣ = ∣ j − k φ + k 5 ∣ ≤ 2 1 + 5 k < 3 k k > c / ( 3 φ 2 ε ) k>c/(3\varphi^2\varepsilon) k > c / ( 3 φ 2 ε ) u τ ≥ k − ε / c > c / ( 3 φ 2 ε ) − ε / c u\tau\ge k-\varepsilon/c>c/(3\varphi^2\varepsilon)-\varepsilon/c uτ ≥ k − ε / c > c / ( 3 φ 2 ε ) − ε / c ε ≤ 1 2 < 2 / 3 / φ = c / ( 2 3 φ ) \varepsilon\le\frac12<\sqrt{2/3}/\varphi=c/(2\sqrt{3}\varphi) ε ≤ 2 1 < 2/3 / φ = c / ( 2 3 φ ) 2 ε / c ≤ c / ( 6 φ 2 ε ) 2\varepsilon/c\le c/(6\varphi^2\varepsilon) 2 ε / c ≤ c / ( 6 φ 2 ε ) c / ( 3 φ 2 ε ) − 2 ε / c ≥ c / ( 6 φ 2 ε ) c/(3\varphi^2\varepsilon)-2\varepsilon/c\ge c/(6\varphi^2\varepsilon) c / ( 3 φ 2 ε ) − 2 ε / c ≥ c / ( 6 φ 2 ε ) ( ε / ( c u ) , ( c / ( 3 φ 2 ε ) − ε / c ) / u ) (\varepsilon/(cu),(c/(3\varphi^2\varepsilon)-\varepsilon/c)/u) ( ε / ( c u ) , ( c / ( 3 φ 2 ε ) − ε / c ) / u ) E ( C S ; ε ) E(C_S;\varepsilon) E ( C S ; ε ) c / ( 6 φ 2 u ε ) = 2 / ( 3 φ 2 u ε ) c/(6\varphi^2u\varepsilon)=\sqrt{2}/(3\varphi^2u\varepsilon) c / ( 6 φ 2 u ε ) = 2 / ( 3 φ 2 u ε ) L ( ε ) ≥ 2 / ( 3 φ 2 u ε ) L(\varepsilon)\ge\sqrt{2}/(3\varphi^2u\varepsilon) L ( ε ) ≥ 2 / ( 3 φ 2 u ε )
Remark 6.12. The order 1 / ε 1/\varepsilon 1/ ε φ \varphi φ { n θ } \{n\theta\} { n θ } [48 ] , and for quasi-uniformity of Kronecker sequences with badly approximable θ \theta θ [21 ] . Upper bounds for the inclusion length via simultaneous Diophantine approximation are due to Naito [40 ] ; Anikushin [3 Theorems 3 and 4] gives a lower bound for the growth exponent of the inclusion length of quasiperiodic functions and notes that both agree for badly approximable exponents. The contribution of Proposition 6.11 is the two-sided inequality for this particular C S C_S C S 0 < ε ≤ 1 2 0<\varepsilon\le\frac12 0 < ε ≤ 2 1
The method (lattice points in a thin strip, and the bounded partial quotients of φ \varphi φ C S C_S C S
Numerical examples
The tabulated numerical values were computed in 60-digit floating-point arithmetic and rounded.
Example 7.1 (u = 1 u=1 u = 1 F = φ 4 F=\varphi^4 F = φ 4 a = φ a=\varphi a = φ b = φ 2 b=\varphi^2 b = φ 2 a + b = φ 3 ≈ 4.236 068 < F = φ 4 ≈ 6.854 102 a+b=\varphi^3\approx4.236\,068<F=\varphi^4\approx6.854\,102 a + b = φ 3 ≈ 4.236 068 < F = φ 4 ≈ 6.854 102 Lemma 2.4 (ii),
S = ( φ 4 , φ 4 − φ , φ 4 − φ 2 , φ 4 − φ 3 ) = ( 3 φ + 2 , 2 φ + 2 , 2 φ + 1 , φ + 1 ) = ( φ 4 , 2 φ 2 , φ 3 , φ 2 ) . S=(\varphi^4,\varphi^4-\varphi,\varphi^4-\varphi^2,\varphi^4-\varphi^3)=(3\varphi+2,2\varphi+2,2\varphi+1,\varphi+1)=(\varphi^4,2\varphi^2,\varphi^3,\varphi^2). S = ( φ 4 , φ 4 − φ , φ 4 − φ 2 , φ 4 − φ 3 ) = ( 3 φ + 2 , 2 φ + 2 , 2 φ + 1 , φ + 1 ) = ( φ 4 , 2 φ 2 , φ 3 , φ 2 ) . numerically ( 6.854 102 , 5.236 068 , 4.236 068 , 2.618 034 ) (6.854\,102,\,5.236\,068,\,4.236\,068,\,2.618\,034) ( 6.854 102 , 5.236 068 , 4.236 068 , 2.618 034 ) Table 1 lists the six differences exactly in Z + Z φ \mathbb{Z}+\mathbb{Z}\varphi Z + Z φ
Table 1. Differences of S S S u = 1 u=1 u = 1 F = φ 4 F=\varphi^4 F = φ 4
Example 7.2 (the Planck scale, u = ν 203 u=\nu_{203} u = ν 203 F = ν 199 F=\nu_{199} F = ν 199 K = K P K=K_{\mathrm{P}} K = K P u = ν 203 u=\nu_{203} u = ν 203 F = ν 199 > ν 200 = a + b F=\nu_{199}>\nu_{200}=a+b F = ν 199 > ν 200 = a + b Lemma 2.7 , ν 199 − ν 200 = ν 201 \nu_{199}-\nu_{200}=\nu_{201} ν 199 − ν 200 = ν 201 ν 199 − ν 201 = ν 200 \nu_{199}-\nu_{201}=\nu_{200} ν 199 − ν 201 = ν 200 ν 199 − ν 202 = ( φ 4 − φ ) u = ( 2 φ + 2 ) u = 2 φ 2 u = 2 ν 201 \nu_{199}-\nu_{202}=(\varphi^4-\varphi)u=(2\varphi+2)u=2\varphi^2u=2\nu_{201} ν 199 − ν 202 = ( φ 4 − φ ) u = ( 2 φ + 2 ) u = 2 φ 2 u = 2 ν 201
S = ( ν 199 , 2 ν 201 , ν 200 , ν 201 ) , S=(\nu_{199},\,2\nu_{201},\,\nu_{200},\,\nu_{201}), S = ( ν 199 , 2 ν 201 , ν 200 , ν 201 ) , and by Corollary 3.3 the differences are ν 202 \nu_{202} ν 202 ν 201 \nu_{201} ν 201 ν 200 \nu_{200} ν 200 ν 203 \nu_{203} ν 203 ν 201 \nu_{201} ν 201 ν 202 \nu_{202} ν 202 Table 2 gives the values. They are computed from the central CODATA values; because of the relative uncertainty 1.1 × 10 − 5 1.1\times10^{-5} 1.1 × 1 0 − 5 K P K_{\mathrm{P}} K P
Table 2. The golden ladder at the Planck scale, ν n = K P φ − n \nu_n=K_{\mathrm{P}}\varphi^{-n} ν n = K P φ − n
Remark 7.3. For u = 1 u=1 u = 1 n = 8 , 12 , 16 n=8,12,16 n = 8 , 12 , 16 ∣ C S ( t + Fib n ) − C S ( t ) ∣ \lvert CS(t+\operatorname{Fib}_n)-CS(t)\rvert ∣ C S ( t + Fib n ) − C S ( t )∣ [ − 50 , 50 ] [-50,50] [ − 50 , 50 ] 12 π φ − n = 0.802 12\pi\varphi^{-n}=0.802 12 π φ − n = 0.802 Proposition 6.5 (ii). This is only a numerical illustration and plays no role in the proofs.
Remark 7.4 (check of Proposition 6.8 ). For u = 1 u=1 u = 1 n ∈ { 1 , 2 , 5 , 10 , 15 } n\in\{1,2,5,10,15\} n ∈ { 1 , 2 , 5 , 10 , 15 } 4 sin ( π φ − n ) μ ( cos π φ − n ) 4\sin(\pi\varphi^{-n})\mu(\cos\pi\varphi^{-n}) 4 sin ( π φ − n ) μ ( cos π φ − n ) Proposition 6.8 (ii) with the maximum of ∣ C S ( t + Fib n ) − C S ( t ) ∣ \lvert CS(t+\operatorname{Fib}_n)-CS(t)\rvert ∣ C S ( t + Fib n ) − C S ( t )∣ 2 ⋅ 10 6 2\cdot10^6 2 ⋅ 1 0 6 t ∈ [ 0 , 2 ⋅ 10 4 ] t\in[0,2\cdot10^4] t ∈ [ 0 , 2 ⋅ 1 0 4 ] 2 ⋅ 10 − 8 2\cdot10^{-8} 2 ⋅ 1 0 − 8 10 − 6 10^{-6} 1 0 − 6 n = 10 n=10 n = 10 μ \mu μ 10 − 15 10^{-15} 1 0 − 15 γ ∈ [ 0 , 1 ] \gamma\in[0,1] γ ∈ [ 0 , 1 ] 6 3 π φ − n 6\sqrt{3}\,\pi\varphi^{-n} 6 3 π φ − n 1.4 × 10 − 8 1.4\times10^{-8} 1.4 × 1 0 − 8 n = 20 n=20 n = 20 R n φ 5 n / π 5 R_n\varphi^{5n}/\pi^5 R n φ 5 n / π 5 n = 2 , 4 , 10 , 20 n=2,4,10,20 n = 2 , 4 , 10 , 20 Proposition 6.8 (iii). This check plays no role in the proofs.
Table 3. Closure residues (in units of u u u x x x Theorem 3 , Proposition 5.3). Fibonacci quotients are given exactly.
Table 4. Explicit almost periods τ n = Fib n \tau_n=\operatorname{Fib}_n τ n = Fib n C S C_S C S u = 1 u=1 u = 1 Proposition 6.5 ).
Relation to the literature: what is classical and what is new
Sidon sets and Golomb rulers
Sidon sets go back to Sidon’s work on trigonometric series [45 ] ; the basic counting bound is due to Erdős and Turán [18 ] , and O’Bryant’s survey [42 ] is a comprehensive guide. Integer Sidon sets are known as Golomb rulers [17 , 23 ] . The golden quadruple is not a Sidon set (Remark 3.4(c)), so it is not a ruler in this sense. There is, however, a formal parallel: a perfect Golomb ruler (Definition 2.2 ) has at most four marks, e.g. { 0 , 1 , 4 , 6 } \{0,1,4,6\} { 0 , 1 , 4 , 6 } { 1 , … , 6 } \{1,\ldots,6\} { 1 , … , 6 } min A = 0 \min A=0 min A = 0 n n n K n K_n K n V ( K n ) → { 0 , 1 , … , ( n 2 ) } V(K_n)\to\{0,1,\ldots,\binom{n}{2}\} V ( K n ) → { 0 , 1 , … , ( 2 n ) } ∣ f ( v ) − f ( w ) ∣ |f(v)-f(w)| ∣ f ( v ) − f ( w ) ∣ K n K_n K n n ≤ 4 n\leq4 n ≤ 4 [23 ] . For the equivalent problem of zero-redundancy linear arrays, Moffet [35 ] credits the construction to Arsac (1955) and the proof that these are the only such arrays to Bracewell (1962); see also [20 §2.4] . Theorems 2 and 2′ exhibit a four-point barrier when the prescribed set of differences is the geometric progression D \mathcal{D} D { c φ n } \{c\varphi^n\} { c φ n } { 1 , … , L } \{1,\ldots,L\} { 1 , … , L }
Difference sets and the size of A − A A-A A − A
Combinatorial difference sets, beginning with Singer’s construction [47 ] , are treated in [6 , 9 ] . For finite A ⊂ R A\subset\mathbb{R} A ⊂ R 2 ∣ A ∣ − 1 ≤ ∣ A − A ∣ ≤ ∣ A ∣ 2 − ∣ A ∣ + 1 2|A|-1\leq|A-A|\leq|A|^2-|A|+1 2∣ A ∣ − 1 ≤ ∣ A − A ∣ ≤ ∣ A ∣ 2 − ∣ A ∣ + 1 [41 , 52 ] . The golden quadruple has ∣ S − S ∣ = 9 |S-S|=9 ∣ S − S ∣ = 9 { 0 , a } + { 0 , b } \{0,a\}+\{0,b\} { 0 , a } + { 0 , b } [26 , 52 ] . Lemma 3.1 is elementary and classical in spirit.
Fibonacci numbers and the golden ratio
The identities used here (the recursion of φ n \varphi^n φ n F i b n + 1 − φ F i b n = ( − φ − 1 ) n \mathrm{Fib}_{n+1}-\varphi\mathrm{Fib}_n=(-\varphi^{-1})^n Fib n + 1 − φ Fib n = ( − φ − 1 ) n [24 , 25 , 56 ] . The quality of the almost periods F i b n / u \mathrm{Fib}_n/u Fib n / u φ \varphi φ ∥ φ F i b n ∥ = φ − n \|\varphi\mathrm{Fib}_n\|=\varphi^{-n} ∥ φ Fib n ∥ = φ − n n ≥ 2 n\ge2 n ≥ 2 [25 , 27 ] ; Proposition 6.8 gives the sharp constant for the specific almost periods F i b n / u \mathrm{Fib}_n/u Fib n / u [49 , 50 , 51 ] : for irrational α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) n ≥ 1 n\ge1 n ≥ 1 { i α } \{i\alpha\} { i α } 0 ≤ i ≤ n 0\le i\le n 0 ≤ i ≤ n R / Z \mathbb{R}/\mathbb{Z} R / Z n + 1 n+1 n + 1 [2 ] : if α = [ 0 ; c 1 , c 2 , … ] \alpha=[0;c_1,c_2,\ldots] α = [ 0 ; c 1 , c 2 , … ] p k / q k p_k/q_k p k / q k η k = ( − 1 ) k ( q k α − p k ) \eta_k=(-1)^k(q_k\alpha-p_k) η k = ( − 1 ) k ( q k α − p k ) n = m q k + q k − 1 + r n=mq_k+q_{k-1}+r n = m q k + q k − 1 + r 1 ≤ m ≤ c k + 1 1\le m\le c_{k+1} 1 ≤ m ≤ c k + 1 0 ≤ r < q k 0\le r<q_k 0 ≤ r < q k η k \eta_k η k η k − 1 − m η k \eta_{k-1}-m\eta_k η k − 1 − m η k η k − 1 − ( m − 1 ) η k \eta_{k-1}-(m-1)\eta_k η k − 1 − ( m − 1 ) η k α = φ − 1 = [ 0 ; 1 , 1 , 1 , … ] \alpha=\varphi^{-1}=[0;1,1,1,\ldots] α = φ − 1 = [ 0 ; 1 , 1 , 1 , … ] c j = 1 c_j=1 c j = 1 m = 1 m=1 m = 1 p k = F i b k p_k=\mathrm{Fib}_k p k = Fib k q k = F i b k + 1 q_k=\mathrm{Fib}_{k+1} q k = Fib k + 1 k ≥ 0 k\ge0 k ≥ 0 φ − 1 = φ − 1 \varphi^{-1}=\varphi-1 φ − 1 = φ − 1 q k α − p k = φ F i b k + 1 − F i b k + 2 = ( − 1 ) k φ − k − 1 q_k\alpha-p_k=\varphi\mathrm{Fib}_{k+1}-\mathrm{Fib}_{k+2}=(-1)^k\varphi^{-k-1} q k α − p k = φ Fib k + 1 − Fib k + 2 = ( − 1 ) k φ − k − 1 η k = φ − k − 1 \eta_k=\varphi^{-k-1} η k = φ − k − 1 η − 1 = 1 \eta_{-1}=1 η − 1 = 1 { φ − k − 2 , φ − k − 1 , φ − k } \{\varphi^{-k-2},\varphi^{-k-1},\varphi^{-k}\} { φ − k − 2 , φ − k − 1 , φ − k } φ − 1 \varphi^{-1} φ − 1 r = q k − 1 r=q_k-1 r = q k − 1 φ \varphi φ
Quasicrystals
Since the discovery of Shechtman, Blech, Gratias and Cahn [44 ] , the golden ratio has been central in the mathematics of aperiodic order: in de Bruijn’s algebraic theory of Penrose tilings [15 ] , in Meyer’s harmonious sets [29 , 33 ] , and in the theory of model sets [4 , 36 ] . The basic one-dimensional example is the Fibonacci chain, generated by the substitution L ↦ L S L\mapsto LS L ↦ L S S ↦ L S\mapsto L S ↦ L φ \varphi φ 1 1 1 [4 ] . Since the second iterate of L L L L S L LSL L S L L S L LSL L S L { 0 , φ , φ 2 , φ 3 } \{0,\varphi,\varphi^2,\varphi^3\} { 0 , φ , φ 2 , φ 3 } φ \varphi φ c = 1 c=1 c = 1 φ n \varphi^n φ n c c c C S C_S C S u Z [ φ ] u\mathbb{Z}[\varphi] u Z [ φ ] φ \varphi φ 1 1 1 Z [ φ ] \mathbb{Z}[\varphi] Z [ φ ] [5 ] [83] and the Fourier module Z [ φ ] / 5 \mathbb{Z}[\varphi]/\sqrt{5} Z [ φ ] / 5 [5 , 8 ] , are modules of this type, namely real multiples of Z [ φ ] \mathbb{Z}[\varphi] Z [ φ ] [4 ] for the general theory of model sets.
Almost periodic functions
Almost periodic functions were introduced by Bohr [10 , 11 ] ; see also [8 , 14 , 28 ] . That trigonometric polynomials are almost periodic is part of the foundations of the theory, and the compactness proof of Lemma 6.3 is standard. The periodicity criterion of Proposition 6.4 is well known; it is a special case of a theorem of Mirotin and Mirotin on sums of continuous periodic functions [34 §2] , and the direct proof given above uses only Lemma 6.1.
Scope: other ratios
Some of the phenomena above are specific to φ \varphi φ ϱ ≈ 1.324 718 \varrho\approx1.324\,718 ϱ ≈ 1.324 718 ϱ 3 = ϱ + 1 \varrho^3=\varrho+1 ϱ 3 = ϱ + 1 [7 , 46 ] ). Then ϱ n + 3 = ϱ n + 1 + ϱ n \varrho^{n+3}=\varrho^{n+1}+\varrho^n ϱ n + 3 = ϱ n + 1 + ϱ n n ∈ Z n\in\mathbb{Z} n ∈ Z ϱ 4 = ϱ 2 + ϱ \varrho^4=\varrho^2+\varrho ϱ 4 = ϱ 2 + ϱ ϱ 5 = ϱ 4 + 1 \varrho^5=\varrho^4+1 ϱ 5 = ϱ 4 + 1
Proposition 8.1. (a) The set R 4 = { 0 , ϱ 2 , ϱ 4 , ϱ 5 } R_4=\{0,\varrho^2,\varrho^4,\varrho^5\} R 4 = { 0 , ϱ 2 , ϱ 4 , ϱ 5 } is a Sidon set whose six differences are exactly 1 , ϱ , ϱ 2 , ϱ 3 , ϱ 4 , ϱ 5 1,\varrho,\varrho^2,\varrho^3,\varrho^4,\varrho^5 1 , ϱ , ϱ 2 , ϱ 3 , ϱ 4 , ϱ 5 each once.
(b) The set P 5 = { 0 , ϱ , ϱ 2 , ϱ 3 , ϱ 4 } P_5=\{0,\varrho,\varrho^2,\varrho^3,\varrho^4\} P 5 = { 0 , ϱ , ϱ 2 , ϱ 3 , ϱ 4 } has all ten positive differences in { ϱ n : n ∈ Z } \{\varrho^n:n\in\mathbb{Z}\} { ϱ n : n ∈ Z } they are ϱ k \varrho^k ϱ k for k = − 3 , − 2 , − 1 , 0 , 1 , 1 , 2 , 2 , 3 , 4 k=-3,-2,-1,0,1,1,2,2,3,4 k = − 3 , − 2 , − 1 , 0 , 1 , 1 , 2 , 2 , 3 , 4
(c) The set R 5 = { 0 , ϱ 2 , ϱ 4 , ϱ 5 , ϱ 7 } = ϱ 7 − ϱ 3 P 5 R_5=\{0,\varrho^2,\varrho^4,\varrho^5,\varrho^7\}=\varrho^7-\varrho^3P_5 R 5 = { 0 , ϱ 2 , ϱ 4 , ϱ 5 , ϱ 7 } = ϱ 7 − ϱ 3 P 5 has all ten positive differences in { ϱ n : n ∈ Z } \{\varrho^n:n\in\mathbb{Z}\} { ϱ n : n ∈ Z } they are ϱ k \varrho^k ϱ k for k = 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 6 , 7 k=0,1,2,3,4,4,5,5,6,7 k = 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 6 , 7
Proof. From ϱ 3 = ϱ + 1 \varrho^3=\varrho+1 ϱ 3 = ϱ + 1 ϱ 4 = ϱ 2 + ϱ \varrho^4=\varrho^2+\varrho ϱ 4 = ϱ 2 + ϱ ϱ 5 = ϱ 3 + ϱ 2 = ϱ 2 + ϱ + 1 = ϱ 4 + 1 \varrho^5=\varrho^3+\varrho^2=\varrho^2+\varrho+1=\varrho^4+1 ϱ 5 = ϱ 3 + ϱ 2 = ϱ 2 + ϱ + 1 = ϱ 4 + 1 ϱ 6 = ϱ 4 + ϱ 3 \varrho^6=\varrho^4+\varrho^3 ϱ 6 = ϱ 4 + ϱ 3 ϱ 7 = ϱ 5 + ϱ 4 \varrho^7=\varrho^5+\varrho^4 ϱ 7 = ϱ 5 + ϱ 4 R 5 R_5 R 5 ϱ 2 , ϱ 4 , ϱ 5 , ϱ 7 \varrho^2,\varrho^4,\varrho^5,\varrho^7 ϱ 2 , ϱ 4 , ϱ 5 , ϱ 7 ϱ 4 − ϱ 2 = ϱ \varrho^4-\varrho^2=\varrho ϱ 4 − ϱ 2 = ϱ ϱ 5 − ϱ 2 = ϱ + 1 = ϱ 3 \varrho^5-\varrho^2=\varrho+1=\varrho^3 ϱ 5 − ϱ 2 = ϱ + 1 = ϱ 3 ϱ 7 − ϱ 2 = ϱ 5 + ϱ 4 − ϱ 2 = ϱ 2 + 2 ϱ + 1 = ϱ 4 + ϱ 3 = ϱ 6 \varrho^7-\varrho^2=\varrho^5+\varrho^4-\varrho^2=\varrho^2+2\varrho+1=\varrho^4+\varrho^3=\varrho^6 ϱ 7 − ϱ 2 = ϱ 5 + ϱ 4 − ϱ 2 = ϱ 2 + 2 ϱ + 1 = ϱ 4 + ϱ 3 = ϱ 6 ϱ 5 − ϱ 4 = 1 \varrho^5-\varrho^4=1 ϱ 5 − ϱ 4 = 1 ϱ 7 − ϱ 4 = ϱ 5 \varrho^7-\varrho^4=\varrho^5 ϱ 7 − ϱ 4 = ϱ 5 ϱ 7 − ϱ 5 = ϱ 4 \varrho^7-\varrho^5=\varrho^4 ϱ 7 − ϱ 5 = ϱ 4 ϱ 7 \varrho^7 ϱ 7 R 4 R_4 R 4 ϱ 2 , ϱ 4 , ϱ 5 , ϱ , ϱ 3 , 1 \varrho^2,\varrho^4,\varrho^5,\varrho,\varrho^3,1 ϱ 2 , ϱ 4 , ϱ 5 , ϱ , ϱ 3 , 1 ϱ > 1 \varrho>1 ϱ > 1 ϱ 7 − ϱ 3 ⋅ { 0 , ϱ , ϱ 2 , ϱ 3 , ϱ 4 } = { ϱ 7 , ϱ 7 − ϱ 4 , ϱ 7 − ϱ 5 , ϱ 7 − ϱ 6 , 0 } = { ϱ 7 , ϱ 5 , ϱ 4 , ϱ 2 , 0 } \varrho^7-\varrho^3\cdot\{0,\varrho,\varrho^2,\varrho^3,\varrho^4\}=\{\varrho^7,\varrho^7-\varrho^4,\varrho^7-\varrho^5,\varrho^7-\varrho^6,0\}=\{\varrho^7,\varrho^5,\varrho^4,\varrho^2,0\} ϱ 7 − ϱ 3 ⋅ { 0 , ϱ , ϱ 2 , ϱ 3 , ϱ 4 } = { ϱ 7 , ϱ 7 − ϱ 4 , ϱ 7 − ϱ 5 , ϱ 7 − ϱ 6 , 0 } = { ϱ 7 , ϱ 5 , ϱ 4 , ϱ 2 , 0 } ϱ 7 − ϱ 6 = ϱ 5 + ϱ 4 − ϱ 4 − ϱ 3 = ϱ 5 − ϱ 3 = ϱ 2 \varrho^7-\varrho^6=\varrho^5+\varrho^4-\varrho^4-\varrho^3=\varrho^5-\varrho^3=\varrho^2 ϱ 7 − ϱ 6 = ϱ 5 + ϱ 4 − ϱ 4 − ϱ 3 = ϱ 5 − ϱ 3 = ϱ 2 P 5 P_5 P 5 R 5 R_5 R 5 y ↦ ϱ − 3 ( ϱ 7 − y ) y\mapsto\varrho^{-3}(\varrho^7-y) y ↦ ϱ − 3 ( ϱ 7 − y ) R 5 R_5 R 5 ϱ − 3 \varrho^{-3} ϱ − 3
What is classical and what is new
Classical or folklore: the set { 0 , φ , φ 2 , φ 3 } \{0,\varphi,\varphi^{2},\varphi^{3}\} { 0 , φ , φ 2 , φ 3 } Elements XIII.8; see Remark 3.2), which up to dilation is also the sequence 0 , 1 , φ , φ 2 0,1,\varphi,\varphi^{2} 0 , 1 , φ , φ 2 [30 (2.4)(b), (3.9)] , together with its difference multiset (Theorem 1, a one-line consequence of φ 2 = φ + 1 \varphi^{2}=\varphi+1 φ 2 = φ + 1 Lemma 2.4 , Lemma 3.1 , Cassini’s identity, the almost periodicity of trigonometric polynomials (Lemma 6.3 ) and the periodicity criterion (Proposition 6.4 ), of which Theorem 4 is a direct consequence.
Known; we include a self-contained elementary proof: the bound of four points, Theorem 2 (i) and Theorem 2 ′(iii), i.e. N ( φ ) = 4 N(\varphi)=4 N ( φ ) = 4 [30 (2.2), (3.9)] , where the maximal sequence 0 , 1 , φ , φ 2 0,1,\varphi,\varphi^{2} 0 , 1 , φ , φ 2 x = φ x=\varphi x = φ [30 (2.4)(b)] , see [30 (3.9)] . Likewise N ( ϱ ) = 5 N(\varrho)=5 N ( ϱ ) = 5 [30 (2.2), (3.10)] (Remark 4.5). Lemma 4.2 and Theorem 2 ′(i) follow from Nagell’s determination of the exceptional units [37 , 38 , 39 ] (Remark 4.3). Version 1 wrongly listed the four-point bound as new.
Explicit geometric formulation, own elementary proof: the uniqueness results Theorem 2 (ii) and Theorem 2 ′(ii). They also follow from Nagell’s list [37 , 38 , 39 ] after normalising an exceptional sequence (cf. [31 ] ), see Remark 4.3; we do not claim them as new.
New (elementary): the rigidity Theorem 3 ′, an explicit geometric rigidity statement with a self-contained elementary proof (it can also be derived from [13 Lemma 3] by a short argument, Remark 5.6; we have not found it in the literature); the sharp asymptotics of Proposition 6.8 (the reduction of the shift norm to the maximum computed in [22 ] , the exact value of E n E_n E n L ( ε ) L(\varepsilon) L ( ε ) 0 < ε ≤ 1 2 0<\varepsilon\le\frac{1}{2} 0 < ε ≤ 2 1 Proposition 6.2 ) and the almost periods Fib n / u \operatorname{Fib}_n/u Fib n / u Proposition 6.5 ) are elementary and serve as their basis; we have not found these statements for C S C_S C S C S ( Fib n / u ) C_S(\operatorname{Fib}_n/u) C S ( Fib n / u )
Supplementary: the closure residues with their Fibonacci values (Theorem 3 , Proposition 5.3), which concern the special quadruples S x S_x S x Theorem 3 ′, and the explicit plastic-number sets (Proposition 8.1 ).
Open questions
Question 1. Let N ( x ) N(x) N ( x ) Definition 4.4 . We know N ( φ ) = 4 N(\varphi)=4 N ( φ ) = 4 N ( ϱ ) = 5 N(\varrho)=5 N ( ϱ ) = 5 [30 (3.9), (3.10)] ; Theorem 2 ′, Proposition 8.1 , Remark 4.5) and N ( 2 ) = 3 N(2)=3 N ( 2 ) = 3 N ( x ) N(x) N ( x ) x > 1 x>1 x > 1 sup x > 1 N ( x ) = 5 \sup_{x>1}N(x)=5 sup x > 1 N ( x ) = 5 N ( x ) ≥ 3 N(x)\ge3 N ( x ) ≥ 3 x l = x k + 1 x^{l}=x^{k}+1 x l = x k + 1 l > k ≥ 0 l>k\ge0 l > k ≥ 0 X l − X k − 1 X^{l}-X^{k}-1 X l − X k − 1 x > 1 x>1 x > 1 x ≥ 1.05 x\ge1.05 x ≥ 1.05 [13 Lemma 3] , who conjecture that their classification of nondegenerate logarithmic lattices (their Theorem 1) holds for all x > 1 x>1 x > 1 λ > 1 \lambda>1 λ > 1 [13 Remark 2] ; the finite list ϱ 1 / m \varrho^{1/m} ϱ 1/ m m ≤ 5 m\le5 m ≤ 5 [13 Lemma 3] belongs to the range x ≥ 1.05 x\ge1.05 x ≥ 1.05 [32 ] and Tverberg [55 ] , preceded by Selmer’s work on X n − X − 1 X^{n}-X-1 X n − X − 1 [43 ] : apart from cyclotomic factors, X n ± X m ± 1 X^{n}\pm X^{m}\pm1 X n ± X m ± 1 [12 Lemma 4.1] for a statement covering exponents with gcd ( n , m ) > 1 \gcd(n,m)>1 g cd( n , m ) > 1 [1 ] classify the common positive roots of X n − X − 1 X^n-X-1 X n − X − 1 X m − X m − 1 − 1 X^m-X^{m-1}-1 X m − X m − 1 − 1 ϱ \varrho ϱ X l − X k − 1 X^l-X^k-1 X l − X k − 1 x > 1 x>1 x > 1 x x x k = 0 k=0 k = 0 X l − 2 X^l-2 X l − 2 x > 1 x>1 x > 1
Question 2. Classify the triples ( x , m , k ) (x,m,k) ( x , m , k ) m m m A ⊂ R A \subset\mathbb{R} A ⊂ R { c , c x , … , c x k } \{c,cx,\ldots,cx^k\} { c , c x , … , c x k } Theorem 3 ′ settles ( m , k ) = ( 4 , 3 ) (m,k)=(4,3) ( m , k ) = ( 4 , 3 ) x = φ x=\varphi x = φ Proposition 8.1 gives examples with ( m , k ) = ( 4 , 5 ) (m,k)=(4,5) ( m , k ) = ( 4 , 5 ) ( 5 , 7 ) (5,7) ( 5 , 7 ) x = ϱ x=\varrho x = ϱ
Question 3. By Proposition 6.11, L ( ε ) ≍ 1 / ε L(\varepsilon)\asymp1/\varepsilon L ( ε ) ≍ 1/ ε lim ε → 0 ε L ( ε ) \lim_{\varepsilon\to0}\varepsilon L(\varepsilon) lim ε → 0 ε L ( ε ) u = 1 u=1 u = 1 6 π φ 3 ≈ 79.8 6\pi\varphi^3\approx79.8 6 π φ 3 ≈ 79.8 2 / ( 3 φ 2 ) ≈ 0.18 2/(3\varphi^2)\approx0.18 2/ ( 3 φ 2 ) ≈ 0.18
Formal verification
The combinatorial and algebraic core of Sections 3–6 is formalised in the proof assistant Lean 4 [16 ] on top of the Mathlib library [53 ] ; Table 5 lists exactly which results are covered. The Bohr almost periodicity of C S C_S C S Lemma 6.3 , Proposition 6.4 , Theorem 4 (b)) and Proposition 8.1 are not formalised. The formalisation is an independent check of part of the proofs given above; the paper is self-contained and does not depend on it.
Table 5. Results of the paper, the corresponding Lean declarations (namespace GoldenDifference), and the status of their formalisation.
Environment and reproduction
Lake project GoldenDifference; all declarations lie in the namespace GoldenDifference. Modules: Basic, Extremality, Rigidity, Algebra, Orbit, Fibonacci, Cosine, AlmostPeriods; audit file Audit.lean.
Source code: to be deposited in a public archive, with a persistent identifier and the commit hash, so that Table 5 can be checked independently; until then it is available from the author on request.
From the root directory of the project, the formalisation is checked with the commands
lake exe cache get # fetch precompiled Mathlib files
lake build # compile and check all modules
lake env lean Audit.lean # print the axioms of the audited declarationsConventions
A finite set A ⊂ R A \subset\mathbb{R} A ⊂ R Finset R. Its positive difference set Δ ( A ) \Delta(A) Δ ( A ) Definition 2.1 ) is posDiff A, the set of all x − y x-y x − y x , y ∈ A x,y\in A x , y ∈ A y < x y<x y < x differenceMultiset A, which has one entry x − y x-y x − y y < x y<x y < x A A A d d d r A ( d ) r_A(d) r A ( d )
cosineSum u t is defined as the right-hand side cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t ) \cos(2\pi ut)+2\cos(2\pi\varphi ut)+2\cos(2\pi\varphi^2ut)+\cos(2\pi\varphi^3ut) cos ( 2 π u t ) + 2 cos ( 2 π φ u t ) + 2 cos ( 2 π φ 2 u t ) + cos ( 2 π φ 3 u t ) C S ( t ) CS(t) C S ( t ) Definition 2.10 , which follows from Theorem 1, is proved on paper and is not formalised. By Remark 3.4(d) this expression depends only on u u u
Formalised statements
Table 5 lists the results of the paper, the corresponding declarations in the namespace GoldenDifference, and the status of each result: formalised (the listed declarations prove the result, with the qualifications stated in the table), partly formalised (only the parts named in the table are proved in Lean), or not formalised .
Table 1 (PDF p. 26).
What is not formalised
The following results are proved by hand in the paper and are not formalised:
Neither are the numerical values of Example 2.8 and § 7 , which are floating-point computations. More generally, no result of the paper is claimed to be formally checked unless it appears in Table 5 as formalised or partly formalised, and in the latter case only to the extent stated there.
Results
In a build carried out on 30 October 2026 in the environment above, lake build terminated without errors, with the message Build completed successfully (2433 jobs). The sources of the project contain no occurrence of sorry, admit, axiom, native_decide or opaque. Hence every declaration of the project, in particular every declaration listed in Table 5 , has been accepted by the Lean kernel with a complete proof, and the project adds no axioms of its own.
The file Audit.lean lists the 27 audited declarations and runs #print axioms on each of them. For every one of these 27 declarations the output contains no axioms other than the three standard axioms of Lean, propext, Classical.choice and Quot.sound; in particular sorryAx does not occur. The remaining eight declarations of Table 5 are not audited directly; each of them is used in the proof of an audited declaration.
Changes in version 2
Priority corrected: the bound of four points in Theorems 2(i) and 2′(iii) (N ( φ ) = 4 N(\varphi)=4 N ( φ ) = 4 N ( ϱ ) ≤ 5 N(\varrho)\leq5 N ( ϱ ) ≤ 5 [30 ] , where the sequence 0 , 1 , φ , φ 2 0,1,\varphi,\varphi^2 0 , 1 , φ , φ 2 [13 Lemma 3] by a short argument (Remark 5.6).
The computational observation on six-point sets for ϱ \varrho ϱ N ( ϱ ) = 5 N(\varrho)=5 N ( ϱ ) = 5 P 5 = { 0 , ϱ , … , ϱ 4 } P_5=\{0,\varrho,\ldots,\varrho^4\} P 5 = { 0 , ϱ , … , ϱ 4 }
References to Nagell [37 ] , Naito [40 ] , Aarts–Fokkink–Kruijtzer, Boyd–Martin–Thom, Selmer, Tverberg [1 , 12 , 43 , 55 ] , Moffet [35 ] and Weiß [57 ] ; the short derivation of Theorem 3′ from [13 Lemma 3] (Remark 5.6).
References to Goldobin–Koelle–Kleiner–Buzdin [22 ] (Proposition 6.8), Anikushin, Goda and Slater [3 , 21 , 48 ] (Proposition 6.11) and Leutbecher–Martinet [31 ] .
Added: Lemma 6.6 (density, replacing the appeal to Kronecker’s theorem), Proposition 6.8 (sharp constants for the almost periods Fib n / u \operatorname{Fib}_n/u Fib n / u L ( ε ) ≍ 1 / ε L(\varepsilon)\asymp1/\varepsilon L ( ε ) ≍ 1/ ε
Acknowledgements
The author used AI tools (Claude, Anthropic; Codex, OpenAI) for drafting and editing the text and for the Lean formalisation. All mathematical content was verified by the author. Theorems 1, 2 and 3′, Lemmas 2.4(iv), 4.2 and 5.2, Theorem 2′(iii), Proposition 5.3, Proposition 6.2 (except the identity for ∣ S ^ ∣ 2 |\widehat{S}|^2 ∣ S ∣ 2 Table 5 (Section 9); the remaining results were verified by hand.
