arXiv:2604.10822v4

Beatty solutions of almost Golomb functional equations

Benoit Cloitre

math.NT39B1211B8311J7137B1068R15

Paper metadata

Original submission: April 12, 2026
Latest revision: July 30, 2026 · version 4
Claimed MSC: 39B12, 11B83, 11J71, 37B10, 68R15
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 43 pages, 2 figures. This version: the shift interval is determined for every non-square r, and the equation is iterated. The Beatty-shift intervals for the triple-nested equation are proved sharp for r=2 and r=3

Abstract

We study the almost Golomb equation of order $r$ , $a\bigl(S(n)\bigr)=n$ , where $S(n)=a(n)+a(n-1)+\cdots+a(n-r+1)$ , for nondecreasing sequences of positive integers. Its greedy solution is $r$ -regular in the sense of Allouche and Shallit. Beyond that one, and for every non-square order $r$ , the equation has another solution, an inhomogeneous Beatty sequence $a(n)=\lfloor n/\sqrt{r}+d \rfloor$ where $d$ is a parameter. No other positive slope occurs, and the equation holds for all $n\ge r$ exactly when $d$ lies in an explicit closed interval depending on $r$ . That interval is a single point when $r=2$ and has positive length for every non-square $r\ge 3$ . Iterating the equation gives, for $k\ge 1$ , $a^{\circ k}\bigl(S(n)\bigr)=a^{\circ(k-1)}(n)$ , with $a^{\circ 0}$ the identity. Its Beatty solutions of positive slope are again of the form $\lfloor n/\sqrt{r}+d \rfloor$ , and the sets of admissible $d$ form an increasing chain. The equation for $k$ holds exactly when $a^{\circ(k-1)}$ agrees at $n$ and at $a(S(n))$ . We determine these solutions for $k=2$ when $r=2$ and $r=3$ . At the right endpoint of the $r=2$ interval the first equation fails on a thin set of indices, which we identify as the return times of an irrational rotation to an explicit interval. The proofs combine equidistribution with an exact computation in $\mathbb{Z}[\sqrt{r}]$ over a finite range of orders.

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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 15, 2026

01Statements4 reported findingsCorrect

The canonical all-order Beatty identity, the complete positive-slope classification for non-square orders, the sharp triple-nested classifications for orders $2$ and $3$ , and the order- $2$ endpoint-defect results are correct under their stated hypotheses. No incorrect or unsupported central statement was found.

Theorems 2 and 3Correct

Canonical solutions and all non-square Beatty shifts

Pages 4–5 and 9–18 · Theorems 2–3, Propositions 8, 12, 20, and 23, and Lemma 24 · arXiv:2604.10822v4

For $a(n)=\lfloor n/\sqrt r+\sqrt r/2\rfloor$ , the sawtooth reduction is exact and the palindromic-walk criterion is equivalent to $a(S(n))=n$ . The analytic bounds cover every sufficiently large non-square order and the exact finite check covers the remaining $5399$ non-square orders $3\leq r\leq5473$ ; the perfect-square calculation separately gives $n$ for odd square roots and $n+1$ for even square roots. For arbitrary shift, density of the irrational orbit and the verified treatment of both boundary phases give exactly the interval $J_r$ . The growth argument also forces the slope $1/\sqrt r$ , so the stated classification is complete.

Full paper, version 4 ↗

Theorems 4 and 32Correct

Sharp triple-nested shift intervals for orders $2$ and $3$

Pages 5 and 19–28 · Theorems 4 and 32 and Sections 4.2–4.6 · arXiv:2604.10822v4

For order $2$ , the defect $a(S(n))-n$ is reduced to the three rotation regimes and is absorbed exactly at repeat positions for $d\in[\sqrt2/2,2(\sqrt2-1)]$ ; orbit density produces failures outside this interval. For order $3$ , the four-case table correctly identifies the repeat status of $n$ and $n-1$ , bounds the defect by $\{-1,0,1\}$ on the proposed interval, and verifies the two absorption conditions. The endpoint orbit exclusions and the density arguments establish both necessity and sufficiency.

Full paper, version 4 ↗

Theorems 5 and 7Correct

Continuous model and forced Beatty slope

Pages 7–8 · Theorems 5 and 7 · arXiv:2604.10822v4

Substitution of an affine function gives exactly the two slopes $c=\pm1/\sqrt r$ and their stated shifts. For a continuous nondecreasing solution, the right inverse is continuous and strictly increasing, which rules out every nontrivial level interval and yields the inverse equation. In the discrete iterated equations, comparison of the linear asymptotics on both sides forces $rc^2=1$ for every $c>0$ .

Full paper, version 4 ↗

Theorems 37 and 40; Proposition 38Correct

Endpoint defect set, return gaps, and substitution structure

Pages 32–37 · Theorem 37, Proposition 38, Lemma 39, and Theorem 40 · arXiv:2604.10822v4

At the right endpoint, direct expansion gives the exact condition $\theta_{n-1}\in[1-c,1/2]$ for a defect. Equidistribution gives its density, and the first-return partition of that interval yields precisely the gaps $3$ , $4$ , and $7$ with the stated frequencies. The affine self-induction maps the return partition according to $3\mapsto34$ , $4\mapsto37$ , and $7\mapsto377$ ; the factor induction and primitivity argument then place the gap word in the asserted minimal substitution subshift.

Full paper, version 4 ↗

02Proofs4 reported findingsCorrect

The proofs of the central results are correct and complete. In particular, the large finite check used in the all-order theorem was independently reconstructed with exact integer comparisons and reproduced the stated unique minimum margin; the analytic large-order argument and the rotation and substitution arguments were also checked in their full stated ranges.

Theorem 2, non-square caseCorrect and complete

Sawtooth reduction, analytic bounds, and finite exact verification

Pages 9–16 and 38–41 · Proposition 12, Lemmas 14–20 and 44, and Appendices A–B · arXiv:2604.10822v4

The jump-point analysis correctly converts the equation to $\operatorname{osc}P\leq\sqrt r-1$ . The block reduction from $P$ to $H$ and then to the length- $s$ rotation sums $E(A)$ preserves every endpoint and includes the truncated final block. Appendix A's four boundary calculations and its middle-range estimate establish the strict inequality for all remaining orders with $\lfloor\sqrt r\rfloor\geq74$ . Independently implementing Appendix B's integer-only comparisons checked all $5399$ non-square orders $3\leq r\leq5473$ : every margin was positive and the unique minimum was $3\sqrt3-5$ at $r=3$ , exactly as stated.

Full paper, version 4 ↗

Theorem 3Correct and complete

Endpoint closure of the shift interval

Pages 16–18 · Proposition 23, Lemma 24, and proof of Theorem 3 · arXiv:2604.10822v4

The arbitrary-shift calculation gives the exact half-open condition $\Phi(\theta_n)\in(\sqrt r(D-1),\sqrt rD]$ . Density supplies necessity and the interior sufficiency. The upper inequality makes the left endpoint admissible, while at the right endpoint the only possible obstruction is an orbit point attaining $\inf\Phi$ ; separating the rational and $\sqrt r$ parts shows every such candidate index is strictly below $r$ . This verifies both closed endpoints without changing the quantified range $n\geq r$ .

Full paper, version 4 ↗

Theorems 4 and 32Correct and complete

Defect absorption and sharpness

Pages 20–28 · Lemmas 28–31 and proof of Theorem 32 · arXiv:2604.10822v4

The local regime decompositions are exhaustive and use the correct repeat thresholds. On each proposed interval the first nesting differs from $n$ by at most one, and the sign of that defect is paired with the required repeated value of $a$ . Outside either endpoint, a nonempty open phase interval violates the corresponding necessary absorption condition, so equidistribution gives infinitely many failures. The exceptional phase at the included endpoint is correctly excluded by irrationality.

Full paper, version 4 ↗

Theorem 40Correct and complete

Exact self-induction of the first-return map

Pages 34–37 · Proposition 38, Lemma 39, Equation (27), and Theorem 40 · arXiv:2604.10822v4

The three return intervals are disjoint with the stated endpoint conventions and exhaust $[1-c,1/2]$ . The orientation-reversing similarity maps them to the correct induced pieces, and the translation identities prove $C(\psi\theta)=\sigma(C(\theta))$ . The finite base-factor check and the shortening bound $\lceil\ell/2\rceil+1<\ell$ for $\ell\geq4$ complete the induction on factors. Since every entry of $M^2$ is positive, the substitution is primitive and its normalized Perron eigenvector agrees with the independently computed return frequencies.

Full paper, version 4 ↗

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Paper: arXiv:2604.10822v4
Authors listed: Benoit Cloitre
Audit date: August 15, 2026

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Beatty solutions of almost Golomb functional equations

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Canonical solutions and all non-square Beatty shifts

Sharp triple-nested shift intervals for orders 222 and 333

Continuous model and forced Beatty slope

Endpoint defect set, return gaps, and substitution structure

Sawtooth reduction, analytic bounds, and finite exact verification

Endpoint closure of the shift interval

Defect absorption and sharpness

Exact self-induction of the first-return map

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Sharp triple-nested shift intervals for orders $2$ and $3$