MathAudit

Paper metadata

Original submission
May 25, 2026
Claimed MSC
11J13
arXiv categories
math.NT
Comments
Short communication, submitted to Fibonacci Quarterly

Abstract

We give a simple proof of a recent result by J. Schleischitz dealing with a counterexample to the uniform Littlewood conjecture. Our construction is based on simple properties of Fibonacci numbers.

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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 20, 2026
01Statements2 reported findingsCorrect

The Fibonacci construction gives real numbers whose uniform Littlewood product has limsup at least 2/(3+5)2/(3+\sqrt5), as stated.

Theorem 1Correct

The uniform Littlewood lower bound follows from the Fibonacci residue construction

Pages 2–3 · Theorem 1 and Sections 4–5 · arXiv:2605.26188v1

For each selected Fibonacci denominator, the coprime residue is chosen so that both normalized coordinates fall in prescribed shrinking intervals. The continued-fraction separation estimate then gives a uniform lower bound for every 1≤x<Fnν1\leq x<F_{n_\nu}, and the approximation errors vanish uniformly as ν→∞\nu\to\infty.

Full paper, version 1 ↗
Section 4Correct

The finite Fibonacci product bound approaches the stated constant

Pages 2–3 · formulas (2)–(4) · arXiv:2605.26188v1

If y/xy/x is not a convergent to Fn−1/FnF_{n-1}/F_n, the standard 1/(2x2)1/(2x^2) separation applies. For a convergent Fk−1/FkF_{k-1}/F_k, the exact continued-fraction error has denominator approaching (3+5)/2(3+\sqrt5)/2. Thus, for every fixed ε>0\varepsilon>0 and all sufficiently large nn, the product is at least 1/(Fn((3+5)/2+ε))1/(F_n((3+\sqrt5)/2+\varepsilon)) throughout 1≤x<Fn1\leq x<F_n; letting ε↓0\varepsilon\downarrow0 gives the theorem's constant.

02Proofs2 reported findingsCorrect

The uniform-distribution lemma, Fibonacci separation estimate, and limiting argument are correct and complete.

Lemma 1Correct and complete

Coprime residues can be localized in both prescribed intervals

Pages 1 and 4 · Lemma 1 and Section 6 · arXiv:2605.26188v1

Uniform distribution first gives an integer a0a_0 whose two normalized coordinates lie strictly inside the target intervals. A nearby Fibonacci number Fk∗F_{k_*} is coprime to FnF_n, and the cited short-shift lemma supplies j=Oσ(Fnσ)j=O_\sigma(F_n^\sigma) for which a=a0+jFk∗a=a_0+jF_{k_*} is coprime to FnF_n. Since k∗=n/2+O(nσ)k_*=n/2+O(n^\sigma), this shift changes both normalized coordinates by o(1)o(1) and keeps them inside the target intervals.

Lemmas 1–2 and Section 5Correct and complete

The finite-level bounds pass uniformly to the constructed limit

Pages 1–4 · auxiliary lemmas and proof of Theorem 1 · arXiv:2605.26188v1

The uniform-distribution and short coprime-shift argument gives a residue in the two prescribed intervals, and the convergent/non-convergent dichotomy supplies the sharp finite Fibonacci estimate. The errors are uniform over the entire minimization range, so taking the limsup is justified.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2605.26188v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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