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The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum — arXiv:2611.00973

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.00973
License (as archived): CC0-1.0
Archived: 2026-11-03T18:49:36.150Z
SHA-256 of the archived text: 86e135d4ca77542443eada4688958b425f7dfd4a3b950605bff28855580b446e
Stable URL of this page: https://metraipolis.com/documento/dc_2e27bb1012d7

How to cite

The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum — arXiv:2611.00973 — arXiv (en), https://arxiv.org/abs/2611.00973 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2e27bb1012d7 (Metraipolis archive dc_2e27bb1012d7, sha256 86e135d4ca77542443eada4688958b425f7dfd4a3b950605bff28855580b446e)

BibTeX

@misc{dc2e27bb1012d7,
  title = {The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum — arXiv:2611.00973},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.00973},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2e27bb1012d7; sha256 86e135d4ca77542443eada4688958b425f7dfd4a3b950605bff28855580b446e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.00973. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $\varphi=(1+\sqrt5)/2$. The four-point set $\{0,\varphi,\varphi^2,\varphi^3\}$ 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 $\varphi^2=\varphi+1$, its six pairwise differences are $1,\varphi,\varphi^2,\varphi^3$, with multiplicities $1,2,2,1$. We write this configuration as the golden quadruple $S=(F, F-a, F-b, F-a-b)$, where $u>0$, $a=\varphi u$, $b=\varphi^2u$ 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 $\{u,\varphi u,\varphi^2u,\varphi^3u\}$ (indeed, not even in the whole set $\{u\varphi^n:n\in\mathbb{Z}\}$), and every four-element set with this property is a translate of $S$. We prove that the golden ratio is forced: $\varphi$ 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 $x>1$, $x\neq\varphi$, 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 $\pm u/F_n^2$ and $\pm F_{n+1}u/F_n^3$. Finally, the cosine sum $C_S(t)=\sum\cos(2\pi d t)$ over the six differences equals $8\cos^2(\pi a t)\cos^2(\pi b t)-2$; 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 $\nu_n=K\varphi^{-n}$ realises the construction with $u=\nu_{203}$, so that the differences are $\nu_{200},\dots,\nu_{203}$; we evaluate it for $K=\sqrt{c^5/(Gh)}$. 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.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO) ; Number Theory (math.NT).
Pagina del lavoro: https://arxiv.org/abs/2611.00973 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T18:15:38.058Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).