arXiv:2603.22045v3

Alternating geometric progressions modulo one and Sturmian words

Qing Lu, Weizhe Zheng

math.NTmath.CO11J5411A6311J7168R15

Paper metadata

Original submission: March 23, 2026
Latest revision: July 23, 2026 · version 3
Claimed MSC: 11J54, 11A63, 11J71, 68R15
arXiv categories: math.NT, math.CO
Paper license: arXiv non-exclusive distribution license ↗
Comments: 13 pages. v3: fixed typos

Abstract

Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $ξ$ such that the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-b)^n)_{n\ge 0}$ is contained in an interval of length $b^{-1}+b^{-2}-b^{-3}$ . In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval.

AI-generated audit

Audit summary

Audited against arXiv v3

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.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The extremal classification for alternating integral geometric progressions, the alternate-order characterization of the doubled Sturmian class, and the more general negative-base word theorem are correct.

Theorem 1.2Correct

Classification at the optimal interval length

Pages 2 and 11 · Theorem 1.2 and its proof · arXiv:2603.22045v3

Lifting the circle interval and setting $v_n=-by_n-y_{n+1}$ produces a bounded aperiodic integer word satisfying $t_{-1/b}(T^nv)=y_n$ . Theorem 1.4 therefore gives $v=-g+w$ with $w\in\mathcal D$ , and telescoping yields the stated representation of $\xi$ after absorbing the harmless integral lift into $g$ . The endpoint and open-interval claims transfer through the same identity. Lemma 4.11 verifies transcendence using the cited theorem for Sturmian expansions.

Full paper, version 3 ↗

Theorem 1.4Correct

Extremal negative-base word classification

Pages 3 and 9–11 · Theorem 1.4 and Section 4 · arXiv:2603.22045v3

The two-letter comparison first forces the alphabet to consist of two adjacent integers. The forbidden-pattern inequalities then force, up to complement, paired zero blocks and hence membership in $\mathcal P$ ; Lemma 4.2 reduces the remaining word to a finite constant prefix followed by a Sturmian word, giving $w\in\mathcal D$ . Conversely, Lemmas 4.9–4.10 put every shift between $100D c_\theta$ and $011D c_\theta$ , whose $t_{-r}$ values differ by $r+r^2-r^3$ . The cited optimal lower bound rules out any shorter interval and makes the endpoint classification exact.

Lu–Zheng, optimal lower bound for negative powers ↗

Corollary 1.3Correct

Alternate-order characterization of $\mathcal D$

Pages 3 and 11–12 · Corollary 1.3 · arXiv:2603.22045v3

For $0<r<1/2$ , Lemma 4.7 identifies alternate order with the numerical order induced by $t_{-r}$ . The bounding words therefore give an interval of exactly $r+r^2-r^3$ , so Theorem 1.4 proves the reverse implication. The prior strict lower bound excludes any strictly tighter lower or upper word and establishes the stated infimum and supremum.

Full paper, version 3 ↗

02Proofs3 reported findingsCorrect

The main proof chain is correct and complete. The order lemmas, forbidden-subword reductions, sharp lower-bound input, and transcendence input all apply with the required hypotheses and scalings.

Lemmas 4.2–4.10Correct and complete

Combinatorial and order-theoretic reductions

Pages 7–9 · Section 4 before Theorem 1.4 · arXiv:2603.22045v3

Lemma 4.2 correctly converts the absence of the paired patterns $01u1$ and $10u0$ into a finite constant prefix followed by a Sturmian word. Lemma 4.4 decomposes $\mathcal D$ into doubled words and their shifts. On each of the three alternate-order regions, formula (4.1) reduces comparison to the ordinary order of equal-slope Sturmian words, so Lemmas 4.8–4.10 supply strict numerical ordering for every $0<r<1$ .

Full paper, version 3 ↗

Proof of Theorem 1.4Correct and complete

Sharp forbidden-pattern estimates and endpoints

Pages 9–11 · Proof of Theorem 1.4 · arXiv:2603.22045v3

Each contradiction uses identity (2.1) with two occurring blocks of equal length, so the interval diameter is multiplied by the correct factor $1+r^k$ . The resulting inequalities successively force adjacent letter values, exclude $010$ and $101$ , control the parity and initial length of zero blocks, and exclude a non-Sturmian residual word. In the converse direction, the previous paper's bound for $S_n(-r)$ scales by $r$ under $t_{-r}(T^nw)=(-r)S_n(-r)$ , giving precisely $r+r^2-r^3$ and validating the no-shorter-interval step.

Lu–Zheng, Theorem 1.2 ↗

Lemma 4.11 and proof of Theorem 1.2Correct and complete

Transcendence and circle lifting

Page 11 · Lemma 4.11 and proof of Theorem 1.2 · arXiv:2603.22045v3

For $w=a^lDs$ , the exact identity expresses $t_{-1/b}(w)$ as an algebraic affine transform of the Sturmian number $t_{1/b^2}(s)$ , so the cited transcendence theorem applies. The lifted recurrence for $y_n$ telescopes absolutely, the finite alphabet is explicit, and irrationality of $y_0$ excludes ultimate periodicity. All hypotheses of Theorem 1.4 are therefore met.

Luca–Ouaknine–Worrell, transcendence of Sturmian expansions ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2603.22045v3
Authors listed: Qing Lu, Weizhe Zheng
Audit date: August 15, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v3
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Alternating geometric progressions modulo one and Sturmian words

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Classification at the optimal interval length

Extremal negative-base word classification

Alternate-order characterization of D\mathcal DD

Combinatorial and order-theoretic reductions

Sharp forbidden-pattern estimates and endpoints

Transcendence and circle lifting

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

Alternate-order characterization of $\mathcal D$