MathAudit

Paper metadata

Original submission
June 15, 2018
Claimed MSC
11J70
arXiv categories
math.NT
Comments
6 pages, in German

Abstract

For real ξξ we consider irrationality measure function ψξ(t)=min⁡1≤q≤t, q∈Z∣∣qξ∣∣ψ_ξ(t) = \min_{1\le q \le t, \, q\in \mathbb{Z}} ||qξ||. We prove that in the case α±β∉Zα\pm β\not\in \mathbb{Z} there exist arbitrary large values of tt with ∣ψα(t)−ψβ(t)∣≥(5+12−1)min⁡(ψα(t),ψβ(t))|ψ_α(t) -ψ_β(t)| \ge\left(\sqrt{\frac{\sqrt{5}+1}{2}}-1\right) \min (ψ_α(t), ψ_β(t)). This result is optimal.

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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 optimal constant (5+1)/2−1\sqrt{(\sqrt5+1)/2}-1 in the comparison of two irrationality-measure functions is correct.

Theorems 1–2Correct

The lower comparison occurs arbitrarily far out and the constant is sharp

Pages 2–6 · Theorems 1–2 and their proofs · arXiv:1806.05989v1

The interlacing of the two convergent-denominator sequences forces one of the adjacent approximation remainders to differ by the golden-ratio threshold. The periodic extremal pattern realizes equality asymptotically, so no larger universal constant can replace it.

Full paper, version 1 ↗
Auxiliary lemmaTypos · no status impact

A minimum and one remainder symbol are typos

Pages 2–3 · Auxiliary Lemma 1 · arXiv:1806.05989v1

Part (iii) requires the maximum of the two displayed quantities, as used in the corollary and proof, not their minimum. In the second symmetric line, the remainder belonging to beta is η\eta, not a second ξ\xi. Both corrections are forced by the stated exchange of alpha and beta.

02Proofs2 reported findingsCorrect

The continued-fraction case analysis and sharpness example are complete after the two auxiliary-lemma symbol corrections.

Proof of Theorem 2Correct and complete

The constructed interlacing pattern proves sharpness

Pages 5–6 · Section 4 and proof of Theorem 2 · arXiv:1806.05989v1

The recursively chosen partial quotients make the two denominator sequences alternate in the prescribed pattern, while their adjacent remainder ratios converge to the golden ratio τ\tau. Consequently the normalized difference of the two irrationality-measure functions approaches τ−1\sqrt{\tau}-1 from the admissible side at every possible comparison point. This rules out any larger universal constant.

Sections 2–4Correct and complete after the stated repairs

The golden-ratio dichotomy and extremal example close

Pages 2–6 · auxiliary lemmas and proofs · arXiv:1806.05989v1

The corrected maximum makes the dichotomy exhaustive, and the recurrence relations translate it directly into the comparison of the two step functions. The constructed interlacing sequence approaches the threshold from the allowed side, proving optimality.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1806.05989v1
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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