arXiv:2607.27780v1

On a question of Gowers related to Littlewood's conjecture

Frederik Broucke, Máté Matolcsi, Szilárd Gy. Révész

math.NT11J1343A35

Paper metadata

Original submission: July 30, 2026
Claimed MSC: 11J13, 43A35
arXiv categories: math.NT
Paper license: CC0 1.0 ↗
Comments: 5 pages

Abstract

In a blogpost in 2009, Gowers raised a possible approach to Littlewood's conjecture in Diophantine approximation, leading to a question about the existence of sufficiently many points in the unit cube such that "hyperbolic distance" between any two of them is large. In this note we answer this question by an explicit construction. This shows that this approach to prove Littlewood's conjecture cannot work, unless some further refinements are added.

AI-generated audit

Audit summary

Audited against arXiv v1

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 central lattice theorem, its finite-point consequence, and the Delsarte bounds are correct. The exact- $n$ conclusion follows immediately from the lattice-point asymptotic by taking a subset and absorbing finitely many small cases, so the mandatory finding gate records no gap there.

Theorem 2Correct

Existence of a product-admissible full-rank lattice

Pages 1–2 · Theorem 2 · arXiv:2607.27780v1

For a totally real number field $K$ of degree $d$ , the Minkowski image of $\mathcal O_K$ is a full-rank lattice in $\mathbb R^d$ . For every nonzero $\alpha\in\mathcal O_K$ , the coordinate product is $|N_{K/\mathbb Q}(\alpha)|$ , a nonzero integer, and is therefore at least $1$ . Hence the lattice meets $H^d=\{x:|x_1\cdots x_d|<1\}$ only at the origin.

Main applicationCorrect

Affirmative answer to Gowers's finite-point question

Page 2 · Paragraph following Remark 3 · arXiv:2607.27780v1

The number of lattice points in the growing cube is asymptotic to $R/\operatorname{vol}(\mathbb R^d/\Lambda)$ , while distinct scaled points have coordinate-product separation at least $1/R$ . For any fixed $c<\operatorname{vol}(\mathbb R^d/\Lambda)^{-1}$ , selecting $n$ points once the count exceeds $n$ , and decreasing $c$ to cover the finitely many smaller values of $n$ , gives the claimed $c/n$ separation for every positive integer $n$ .

Section 3Correct

Delsarte quantities and asymptotic bounds

Pages 3–4 · Section 3 · arXiv:2607.27780v1

The autocorrelation measure of a separated point set has the stated support, atom, mass, and nonnegative Fourier coefficients. The test function $\varphi=\varepsilon^{-1}\chi*\chi$ is positive definite, is nonpositive outside $H_\varepsilon^d$ because it vanishes there, and has integral $\varepsilon$ . These facts, together with the applicable strong-duality theorem, give the displayed bounds and $N_d(\varepsilon)\asymp\varepsilon^{-1}$ .

Strong duality source, Theorem 5.6 ↗

02Proofs4 reported findingsCorrect

All substantive proof steps and material cited inputs are correct and complete after immediate-consequence closure. Three literal notation defects have unique, verified repairs and remain visible in yellow without lowering the proof status.

Littlewood lower boundTypo

An extra closing parenthesis should be removed

Page 1 · Paragraph immediately after the display defining $X_n$ · arXiv:2607.27780v1

The factor is printed as $\lVert(l-k))\alpha\rVert$ . Remove the extra closing parenthesis to obtain $\lVert(l-k)\alpha\rVert$ . The correction is uniquely mechanical and changes no inequality or argument.

Footnote 1Typo

The cyclotomic subgroup generator has the wrong exponent

Page 2 · Footnote 1 to the proof of Theorem 2 · arXiv:2607.27780v1

The printed automorphism $\zeta_p\mapsto\zeta_p^d$ need not generate a subgroup of order $(p-1)/d$ ; for $d=2$ and $p=5$ , exponent $2$ generates the whole Galois group. If $g$ is a primitive root modulo $p$ , replace the subgroup by $H=\langle\zeta_p\mapsto\zeta_p^{g^d}\rangle$ , equivalently the unique subgroup of order $(p-1)/d$ . Its fixed field has degree $d$ , and it is totally real because $(p-1)/d$ is even, so $H$ contains complex conjugation. The required subgroup order and index uniquely determine this verified mechanical repair; the norm proof is unchanged.

Scaling displayTypo

The strict inequality should be non-strict

Page 2 · Display immediately after the asymptotic for $\#Y_R$ · arXiv:2607.27780v1

The display prints $\left|\prod_{i=1}^d(\lambda_i-\lambda_i')/R^{1/d}\right|>1/R$ . Replace $>$ by $\geq$ . Theorem 2 gives $\left|\prod_i(\lambda_i-\lambda_i')\right|\geq1$ , with equality possible, and this non-strict bound is exactly what the application requires.

Section 3Correct and complete

Convolution and strong-duality argument

Pages 3–4 · Section 3 · arXiv:2607.27780v1

The Fourier-positivity, support, normalization, and mass calculations are correct with the stated Haar normalization. The cited strong-duality result applies to this compact Abelian-group setting, and the paper also gives a valid direct proof of the upper bound $D'(H_\varepsilon^d)\leq1/\varepsilon$ .

Strong duality source, Theorem 5.6 and Remark 5.4 ↗

03Novelty1 reported findingContains non-new main statements

The central admissible-lattice theorem and the resulting finite-point construction follow directly from a publicly available 2018 Frolov-lattice construction.

Theorem 2 and main applicationPreviously established

The admissible lattice and resulting point construction were already available

Theorem 2 on pages 1–2 and its consequence for Problem 1 on page 2 · arXiv:2607.27780v1

Kacwin, Oettershagen, Mario Ullrich, and Tino Ullrich, public on February 23, 2018, define an admissible lattice by $\inf_{k\neq0}|\prod_i(Vk)_i|>0$ . Their Proposition 2.2 and Equations (1.5)–(1.6) construct, in every dimension, a Vandermonde lattice with minimum product $1$ . Thus every nonzero $\lambda\in V\mathbb Z^d$ satisfies $|\prod_i\lambda_i|\geq1$ , exactly the condition $\Lambda\cap H^d=\{0\}$ in Theorem 2. Their scaling and lattice-point formulas, Equations (1.3)–(1.4), then immediately yield point sets in the unit cube with cardinality proportional to the scale and product separation proportional to its reciprocal; selecting exactly $n$ points gives the stated answer to Gowers's question.

Kacwin–Oettershagen–Ullrich–Ullrich, Definition 2.1 and Proposition 2.2 ↗

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:2607.27780v1
Authors listed: Frederik Broucke, Máté Matolcsi, Szilárd Gy. Révész
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 v1
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.

On a question of Gowers related to Littlewood's conjecture

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Existence of a product-admissible full-rank lattice

Affirmative answer to Gowers's finite-point question

Delsarte quantities and asymptotic bounds

An extra closing parenthesis should be removed

The cyclotomic subgroup generator has the wrong exponent

The strict inequality should be non-strict

Convolution and strong-duality argument

The admissible lattice and resulting point construction were already available

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match