MathAudit

Paper metadata

Original submission
November 18, 2019
Claimed MSC
arXiv categories
math.NT, math.CO
Comments
14 pages

Abstract

We prove that for any prime pp there is a divisible by pp number q=O(p30)q = O(p^{30}) such that for a certain positive integer aa coprime with qq the ratio a/qa/q has bounded partial quotients. In the other direction we show that there is an absolute constant C>0C>0 such that for any prime pp exist divisible by pp number q=O(pC)q = O(p^{C}) and a number aa, aa coprime with qq such that all partial quotients of the ratio a/qa/q are bounded by two.

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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
01Statements3 reported findingsCorrect

The modular forms of Zaremba's conjecture, with a bounded digit alphabet and with digits bounded by two, are correct.

Theorem 10Correct and complete

Product growth forces an intersection with the Borel subgroup

Pages 6–10 · Lemma 8, Corollary 9, and Theorem 10 · arXiv:1911.07487v1

The Bruhat decomposition identifies every non-Borel double coset, and the representation estimate bounds the multiplicative energy between the product set and the Borel subgroup. If the intersection were empty, the resulting growth inequality would contradict the assumed size once the bounded product power is chosen. This supplies the group-theoretic step used in both main continued-fraction constructions.

Theorems 1–2Correct

A denominator divisible by every prescribed prime is obtained with polynomial size

Pages 1–2 and 10–13 · Theorems 1–2 and Section 5 · arXiv:1911.07487v1

Finite continued fractions are encoded by products in SL⁡2(Fp)\operatorname{SL}_2(\mathbb F_p). The product-growth and Borel-subgroup intersection bounds force a bounded power of the continued-fraction set to meet the required subgroup, making the resulting denominator divisible by pp while keeping it polynomial in pp.

Full paper, version 1 ↗
Theorem 2 specializationCorrect

The digit-two construction uses the same verified group mechanism

Pages 8–13 · Theorem 14 and conclusion of Section 5 · arXiv:1911.07487v1

Hensley's cardinality estimate gives a generating set of fixed positive power size even for digit bound two. Repeated product growth reaches the Borel subgroup in a bounded number of steps, which yields an absolute polynomial exponent CC.

02Proofs1 reported findingCorrect

The representation-theoretic, product-growth, trace-counting, and continued-fraction arguments are correct and complete.

Sections 3–5Correct and complete

The group estimates imply the modular continued-fraction conclusions

Pages 3–13 · Lemmas 4–19 and proofs of Theorems 1–2 · arXiv:1911.07487v1

The Fourier transform of a Borel subgroup is computed with the correct representation dimensions, the Helfgott-type growth dichotomy supplies bounded product length, and the trace multiplicity estimate prevents concentration. The matrix-continuant identity then converts subgroup membership into the desired divisible denominator.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1911.07487v1
Authors listed
Nikolay G. Moshchevitin, Ilya D. Shkredov
Audit date
August 20, 2026
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