MathAudit

Paper metadata

Original submission
February 21, 2012
Latest revision
February 22, 2012 · version 2
Claimed MSC
11A55
arXiv categories
math.NT
Comments
Submitted to the proceedings of the Palanga Conference in Number Theory (September 2011)

Abstract

We study some properties of the function μα(t)μ_α(t) associated with the Minkowski diagonal continued fraction for real αα.

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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
01Statements2 reported findingsCorrect

The formula for the Minkowski diagonal invariant, its spectrum endpoints, and the quadratic-irrational oscillation theorem are correct.

Theorems 1 and 2Correct

Continued-fraction formula and spectrum endpoints

Pages 3-4 and 5-8 · Theorems 1-2 · arXiv:1202.4622v2

Each adjacent pair of retained convergents is either (qν−1,qν+1)(q_{\nu-1},q_{\nu+1}) with aν+1=1a_{\nu+1}=1 or (qν,qν+1)(q_\nu,q_{\nu+1}). Lemmas 3-8 compute the maximum of tμα(t)t\mu_\alpha(t) on the corresponding segment as GG or FF. Their ranges give 1/4≤m(α)≤1/21/4\leq m(\alpha)\leq1/2, and the two continued-fraction examples attain both endpoints.

Full paper, version 2 ↗
Theorem 3Correct

Quadratic-irrational difference functions oscillate

Pages 5 and 8-9 · Theorem 3 and Section 10 · arXiv:1202.4622v2

Distinct quadratic fields make the logarithms of fundamental units rationally independent. Kronecker approximation aligns subsequences where tμβ(t)t\mu_\beta(t) approaches either λ(β)\lambda(\beta) or m(β)m(\beta) with one where tμα(t)t\mu_\alpha(t) approaches λ(α)\lambda(\alpha). Hypothesis (17) then forces opposite signs along the two aligned subsequences.

02Proofs2 reported findingsCorrect

The continued-fraction identities, hyperbolic-rotation maxima, and unit-alignment argument are correct; several variable names in Section 10 are harmless typos.

Lemmas 2-8Correct and complete

The segment maxima are derived correctly

Pages 5-8 · Lemmas 2-8 · arXiv:1202.4622v2

The standard identities for qνξνq_\nu\xi_\nu and ξν/ξν+1\xi_\nu/\xi_{\nu+1} give the stated normalizing factors. The diagonal maps preserve tμt\mu and make each relevant segment perpendicular to the diagonal; Lemmas 5-6 place its endpoints on opposite sides, so the maximum is attained at the diagonal and equals GG or FF.

Section 10 notationTypos · no status impact

Three labels in the oscillation proof are typographical errors

Pages 8-9 · proof of Theorem 3 · arXiv:1202.4622v2

The second field is Q(β)\mathbb Q(\beta), not a second Q(α)\mathbb Q(\alpha); the two target phases are ω1,ω2\omega_1,\omega_2, not ω1,ω1\omega_1,\omega_1; and the second index pair is (ν2,n,κ2,n)(\nu_{2,n},\kappa_{2,n}). The parallel formulas immediately determine all three corrections.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1202.4622v2
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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