MathAudit

Paper metadata

Original submission
March 8, 2022
Latest revision
January 17, 2024 · version 2
Claimed MSC
arXiv categories
math.NT
Paper license
CC BY-NC-ND 4.0 ↗
Comments
15 pages, 2 figures

Abstract

Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation, optimal in terms of this geometry.

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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 20, 2026
01Statements3 reported findingsCorrect

The lower bound for the dimension of spans of best simultaneous approximations in three dimensions and the realizing constructions are correct.

Proof of Theorem 1Correct and complete

The exponent root follows from the determinant alternatives

Pages 5–8 · Lemma 2 and proof of Theorem 1 · arXiv:2203.04242v2

The two determinant alternatives give the displayed inequalities between consecutive height and error scales. Substituting a uniform exponent majorant reduces them to the polynomial inequality defining gk(λ)g_k(\lambda) in Lemma 1. Monotonicity selects its unique root at least one, and the resulting bound is exactly the conclusion of Theorem 1.

Theorems 1–2Correct

The geometric lower bound is sharp for every fixed block parameter

Pages 2–5 and 8–18 · Theorems 1–2 and their proofs · arXiv:2203.04242v2

Minkowski's theorem and the determinant estimates force sufficiently many consecutive best-approximation vectors to span the claimed dimension. The staged construction then prescribes successive primitive vectors and error scales so that no competitor enters the intervening cylinders, attaining equality.

Full paper, version 2 ↗
Lemmas 2–3Typos · no status impact

A denominator subscript and two translated-point symbols are typos

Pages 6–10 · Lemmas 2–3 · arXiv:2203.04242v2

The induction denominator is θk−σSk\theta^k-\sigma S_k, not θk−σSk\theta^k-\sigma S^k. The points denoted XjX_j in Lemma 3 are translates of ZjZ_j by the displayed scaled directions; the two definitions omit the leading Zj+Z_j+. The subsequent distance calculations use the corrected forms.

02Proofs1 reported findingCorrect

The determinant lower bound and the recursive sharpness construction are correct and complete after the displayed symbol repairs; the higher-block stages genuinely repeat the verified first stage with the listed parameter substitutions.

Sections 3–5Correct and complete after the stated repairs

The no-competitor cylinders persist through the induction

Pages 5–20 · geometric lemmas and proof of Theorem 2 · arXiv:2203.04242v2

At each stage the translated simplex contains exactly the designated primitive points, the determinant bounds exclude all other lattice points at the relevant height, and the chosen scale separation preserves every earlier best approximation. The stated higher-kk substitutions satisfy the same inequalities.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2203.04242v2
Authors listed
Antoine Marnat, Nikolay Moshchevitin
Audit date
August 20, 2026
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