MathAudit

Paper metadata

Original submission
May 30, 2022
Claimed MSC
11J20
arXiv categories
math.NT
Comments
in German

Abstract

We discuss some easy statements dealing with linear inhomogeneous Diophantine approximation. Surprisingly, we did not find some of them in the literature.

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 20, 2026
01Statements3 reported findingsCorrect

The inhomogeneous approximation statements, including the primitive-point conclusions in Theorem 7, are correct after two local parameter and index corrections.

Proof of Theorem 3Correct and complete

Jarnik's growth estimate and transference give the claimed approximant

Pages 2–3 · proof of Theorem 3 · arXiv:2205.14961v1

Jarnik's estimate provides tν→∞t_\nu\to\infty with tνψΘ⊤(tν/(2Kε))→∞t_\nu\psi_{\Theta^\top}(t_\nu/(2K\varepsilon))\to\infty. Applying Theorem 1 with C=ε/tνC=\varepsilon/t_\nu and X=(KψΘ⊤(tν/(2Kε)))−1X=(K\psi_{\Theta^\top}(t_\nu/(2K\varepsilon)))^{-1} gives an integer vector with error below ε/tν\varepsilon/t_\nu and ∣x∣≤X≤tν|\boldsymbol x|\leq X\leq t_\nu for large ν\nu. Hence the error is below ε/∣x∣\varepsilon/|\boldsymbol x|, exactly as Theorem 3 claims.

Theorems 3 and 7Correct

The primitive inhomogeneous approximants exist with the stated quality

Pages 2–3 and 7–8 · Theorems 3 and 7 · arXiv:2205.14961v1

The transference parallelepiped supplies a lattice point at the required approximation scale. Adding a short multiple of a basis vector makes its coefficients coprime; the perturbation is smaller than the available inhomogeneous error in both parameter regimes.

Full paper, version 1 ↗
Theorem 3 and proof of Theorem 7Minor formal corrections · no status impact

A variable range and one endpoint exponent need local correction

Pages 2 and 7 · Theorem 3 and final proof · arXiv:2205.14961v1

The independent indices in Theorem 3 satisfy 1≤i<j≤m1\leq i<j\leq m, not j≤nj\leq n. In the final perturbation argument choose any δ<(m−1)/((m+1)(m+3))\delta<(m-1)/((m+1)(m+3)) rather than equality; then the residual factor tends to zero exactly as required. The primitive-point lemma permits every positive δ\delta, so this change has no downstream effect.

02Proofs1 reported findingCorrect

The transference and primitive-perturbation proofs are complete after choosing the final exponent strictly inside its admissible range.

Proof of Theorem 7Correct and complete after the stated repair

A strict interior exponent makes the final error vanish

Pages 7–8 · equations (20)–(21) · arXiv:2205.14961v1

The coefficient bound is polynomial in the transference parameter, and the coprimality adjustment grows by an arbitrarily small power. With the strict choice of δ\delta, its contribution is o(1/T1)o(1/T_1) and the stated infinitely many primitive approximants follow.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2205.14961v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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