arXiv:2506.01736v2

The bad and rough rotation is Poissonian

Manuel Hauke

math.NT11J7111N2511N8011K4511K6011J83

Paper metadata

Original submission: June 2, 2025
Latest revision: August 14, 2026 · version 2
Claimed MSC: 11J71, 11N25, 11N80, 11K45, 11K60, 11J83
arXiv categories: math.NT
Paper license: CC BY-NC-SA 4.0 ↗
Comments: 35 pages; added some references and an appendix concerning numerical experiments

Abstract

Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences $(a_nα)_{n \in \mathbb{N}} \mod 1$ with $(a_n)_{n \in \mathbb{N}} \subseteq \mathbb{N}$ and $α$ irrational have been extensively studied in the last decades. In this article, we prove that for $(a_n)_{n \in \mathbb{N}}$ arising from the set of rough numbers with explicit roughness parameters and any badly approximable $α$ , $(a_nα)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence $(a_nα)_{n \in \mathbb{N}} \mod 1$ with these properties. Further, we show that this result is false for Lebesgue almost every $α$ , thereby disproving a conjecture of Larcher and Stockinger [Math. Proc. Camb. Phil. Soc. 2020]. The method of proof makes use of an equidistribution result mod $d$ in diophantine Bohr sets which might be of independent interest.

AI-generated audit

Audit summary

Audited against arXiv v2

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 18, 2026

01Statements3 reported findingsContains unsupported statements

The triangular-array Poissonian-correlation theorem and the Bohr-set equidistribution theorem are supported. The corresponding claims for the single sequence of variably rough numbers, including the sequence clause of Theorem 2, are not verified under the printed hypotheses: the passage from triangular arrays to that sequence silently needs monotonicity and slow variation of $f(x)=x^{r(x)}$ , neither of which follows from monotone decrease of $r$ .

Theorem 1, triangular-array clauseCorrect

Poissonian correlations and gaps for uniformly rough triangular arrays

Pages 3 and 19–26 · Theorem 1 and Sections 5.1–5.2 · arXiv:2506.01736v2

For fixed order $k$ , the sieve estimate reduces the correlation count to an average of the singular-series factor over Diophantine Bohr sets. Theorem 3 and the Ostrowski-cylinder argument give the required uniform residue-class distribution, Lemmas 15–16 control the large-prime tail and average the local factors, and the step-function partition converges to the required volume. After the local repair recorded in Part 2 for one overstrong intermediate growth assertion, these steps establish Poissonian correlations of every fixed order and hence the stated gap law for the triangular arrays.

Theorem 3Correct

Equidistribution modulo a fixed integer in shrinking Diophantine Bohr sets

Pages 6 and 13–19 · Theorem 3 and its proof · arXiv:2506.01736v2

The Ostrowski expansion reduces each one-sided shrinking interval to finitely many cylinder sets. On each cylinder, separated two-digit blocks provide independent residue increments whose support contains $0$ and two residues generating $\mathbb Z/d\mathbb Z$ because consecutive convergent denominators are coprime. The resulting finite-state random walk converges uniformly to the uniform distribution, and the cylinder approximation error tends to zero. The reversed consecutive-error brackets and the block-index slips recorded in Part 2 have unique local corrections and do not change this argument.

Theorems 1 and 2, sequence clausesNot able to verify

The triangular-array results are not transferred to the stated variable-threshold sequence

Pages 3–4 and 26–28 · Theorems 1–2, Section 5.3, and opening of Section 6 · arXiv:2506.01736v2

The sequence in Definition 1 contains $n$ according to the threshold $f(n)$ , whereas the proved triangular-array results use one fixed threshold $f(x)$ for all $n\leq x$ . Section 5.3 sets $z=f(x)$ and $z^-=f(x/(\log x)^{k+1})$ and uses both $\Phi(x,z)\leq N(x,f)\leq\Phi(x,z^-)$ and $\log z/\log z^-\to1$ . The hypotheses say only that $r(x)$ decreases to zero and that $r(x)\gg_A(\log\log x)^A/\log x$ ; they do not imply that $f(x)=x^{r(x)}$ is increasing or that $r(x/(\log x)^{k+1})/r(x)\to1$ . For example, take $X_j=\exp(\exp(j^2))$ and $r(x)=2^{-j}$ on $[X_j,X_{j+1})$ . This $r$ satisfies every printed lower bound, but at $x=X_j$ the two logarithmic thresholds have asymptotic ratio $1/2$ , not $1$ , and $f$ has a downward jump. Thus the sandwich used for Theorem 1 is unavailable. Section 6 transfers Theorem 2 to the sequence only by citing the same argument. No counterexample to the two sequence conclusions is established, but no replacement proof under the printed scope is supplied.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The triangular-array and Bohr-set arguments are substantially complete, but the sequence-conversion proof uses unproved monotonicity and slow variation and therefore does not cover Theorems 1–2 as stated. Lemma 16 also contains a false growth inference, although the estimate actually needed there has a verified repair. Several Ostrowski-index slips are uniquely correctable and harmless.

Section 5.3Incomplete as written

The threshold sandwich assumes properties of $f$ absent from Theorems 1–2

Pages 26–28 · Section 5.3 and first paragraph of Section 6 · arXiv:2506.01736v2

For every $n$ in the retained range, the displayed comparison requires $f(x/(\log x)^{k+1})\leq f(n)\leq f(x)$ , and the normalization requires $\Phi(x,z^-)/\Phi(x,z)\to1$ , equivalently here $\log z/\log z^-\to1$ . Monotonicity of $r$ gives neither assertion because the factor $\log x$ moves in the opposite direction and $r$ may have large relative downward jumps. The explicit step function in Part 1 demonstrates that the claimed ratio and sandwich can fail within the stated hypotheses. Downstream dependency: the single-sequence conclusion of Theorem 1 and the single-sequence and almost-everywhere clauses of Theorem 2. Repair classification: No repair supplied. One possible route is to add hypotheses making $f$ eventually increasing and slowly varying on polylogarithmic scales; retaining the present scope would require a new comparison argument.

Lemma 16, equation (48)Incorrect as written

An unnecessarily strong lower bound for the Rankin parameter is false

Page 23 · proof of Lemma 16 immediately before equation (48) · arXiv:2506.01736v2

The proof defines $U=\exp\!\left(\frac{\log T}{\log r_2}\log2\right),\qquad r_2=(\log\log\log x)^{C'},$ assumes $T\gg_A(\log\log x)^A$ for every fixed $A$ , and then claims $U\gg_A(\log\log x)^A$ for every $A$ . This implication is false: with $L=\log\log x$ and $T=L^{\sqrt{\log\log L}}$ , the hypothesis holds for every fixed $A$ , while $U=L^{o(1)}$ . Repair classification: Verified repair. The argument needs only $(r_2\log r_2)^K=o(U)$ for the fixed exponent $K=2^{c_k+1}$ . The hypothesis gives $\log T/\log L\to\infty$ ; hence $\log U\gg (\log L)/\log\log L$ , whereas $\log((r_2\log r_2)^K)=O(\log\log L)$ . The needed comparison and therefore equation (48) follow without the false stronger claim.

Proposition 13 and proof of Theorem 3Minor formal correction

The consecutive approximation errors are bracketed in the impossible order

Pages 14 and 18 · Proposition 13 and first line of the proof of Theorem 3 · arXiv:2506.01736v2

The paper prints $|\delta_m|\leq\gamma<|\delta_{m+1}|$ in Proposition 13 and $|\delta_{m-1}|<\rho_x\leq|\delta_m|$ in the proof of Theorem 3. Since the convergent errors $|\delta_i|$ decrease, both intervals are empty. Reverse the brackets consistently, for example $|\delta_{m+1}|<\gamma\leq|\delta_m|$ , and make the corresponding one-index relabeling in Theorem 3. The cylinder proof already uses only the scale being between consecutive errors, so this local correction changes no estimate or conclusion.

Lemma 14, two-digit blocksTypo

Three swapped indices obscure the independent-block construction

Pages 16–18 · items (ii)–(iii) and their proof in Lemma 14 · arXiv:2506.01736v2

For the conditioned pair $(b_i,b_{i+1})$ with $b_{i-1}=b_{i+2}=0$ , the exceptional admissible pair must be $(0,a_{i+2})$ , not $(a_{i+2},0)$ ; the bijection must retain $b_{i-1}=b_{i+2}=0$ , not $b_{i-1}=b_{i+1}=0$ ; and the integer contribution is $xq_i+yq_{i+1}$ , not $xq_{i+1}+yq_i$ . The later construction fixes $b_{4i}=b_{4i+3}=0$ around $(b_{4i+1},b_{4i+2})$ and therefore determines all three corrections uniquely. With them, the uniformity and independence argument is unchanged.

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:2506.01736v2
Authors listed: Manuel Hauke
Audit date: August 18, 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 v2
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.

The bad and rough rotation is Poissonian

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Poissonian correlations and gaps for uniformly rough triangular arrays

Equidistribution modulo a fixed integer in shrinking Diophantine Bohr sets

The triangular-array results are not transferred to the stated variable-threshold sequence

The threshold sandwich assumes properties of fff absent from Theorems 1–2

An unnecessarily strong lower bound for the Rankin parameter is false

The consecutive approximation errors are bracketed in the impossible order

Three swapped indices obscure the independent-block construction

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The threshold sandwich assumes properties of $f$ absent from Theorems 1–2