The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum
Abstract
Let φ=(1+√5)/2. The four-point set {0,φ,φ2,φ3} is classical: up to a dilation it consists of the two endpoints and the two crossing points on a side of the regular pentagram (Euclid, Elements XIII.8), and, by the identity φ2=φ+1, its six pairwise differences are 1,φ,φ2,φ3, with multiplicities $1,2,2,1$. We write this configuration as the golden quadruple S=(F, F−a, F−b, F−a−b), where u>0, a=φu, b=φ2u and F>a+b, and study its extremal and rigidity properties. We prove that it is maximal and unique: no five distinct real numbers have all their positive differences in {u,φu,φ2u,φ3u} (indeed, not even in the whole set {uφn:n∈ℤ}), and every four-element set with this property is a translate of S. We prove that the golden ratio is forced: φ is the only ratio x>1 for which some four-point set of reals has the four-term progression {c,cx,cx2,cx3} as its set of distinct differences. For the analogous quadruple with base x>1, x≠φ, the failure is measured by two closure residues, (x2−x−1)u and x(x2−x−1)u, which along Fibonacci quotients equal ± u/Fn2 and ± Fn+1u/Fn3. Finally, the cosine sum CS(t)=∑cos(2πd t) over the six differences equals 8cos2(πa t)cos2(πb t)−2; it is not periodic, but it is almost periodic in the sense of Bohr, with explicit almost periods Fn/u. For an arbitrary scale K>0 the golden ladder νn=Kφ−n realises the construction with u=ν203, so that the differences are ν200,…,ν203; we evaluate it for K=√(c5/(Gh)). We state explicitly what is classical and what is new, in relation to difference sets, Golomb rulers, Sidon sets, Fibonacci numbers, quasicrystals and almost periodic functions.
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BibTeX
@misc{przybylski2026golden,
title = {{The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum}},
author = {Michał Przybylski},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00973},
eprint = {2611.00973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00973},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum
AU - Michał Przybylski
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00973
UR - https://arxiv.org/abs/2611.00973
AB - Let φ=(1+√5)/2. The four-point set {0,φ,φ^2,φ^3} is classical: up to a dilation it consists of the two endpoints and the two crossing points on a side of the regular pentagram (Euclid, Elements XIII.8), and, by the identity φ^2=φ+1, its six pairwise differences are 1,φ,φ^2,φ^3, with multiplicities $1,2,2,1$. We write this configuration as the golden quadruple S=(F, F−a, F−b, F−a−b), where u>0, a=φu, b=φ^2u and F>a+b, and study its extremal and rigidity properties. We prove that it is maximal and unique: no five distinct real numbers have all their positive differences in {u,φu,φ^2u,φ^3u} (indeed, not even in the whole set {uφ^n:n∈ℤ}), and every four-element set with this property is a translate of S. We prove that the golden ratio is forced: φ is the only ratio x>1 for which some four-point set of reals has the four-term progression {c,cx,cx^2,cx^3} as its set of distinct differences. For the analogous quadruple with base x>1, x≠φ, the failure is measured by two closure residues, (x^2−x−1)u and x(x^2−x−1)u, which along Fibonacci quotients equal ± u/F_n^2 and ± F_n+1u/F_n^3. Finally, the cosine sum C_S(t)=∑cos(2πd t) over the six differences equals 8cos^2(πa t)cos^2(πb t)−2; it is not periodic, but it is almost periodic in the sense of Bohr, with explicit almost periods F_n/u. For an arbitrary scale K>0 the golden ladder ν_n=Kφ^−n realises the construction with u=ν_203, so that the differences are ν_200,…,ν_203; we evaluate it for K=√(c^5/(Gh)). We state explicitly what is classical and what is new, in relation to difference sets, Golomb rulers, Sidon sets, Fibonacci numbers, quasicrystals and almost periodic functions.
ER -