MathAudit

Paper metadata

Original submission
August 19, 2021
Latest revision
November 27, 2021 · version 2
Claimed MSC
11J13
arXiv categories
math.NT
Comments
in German, this version contains minor corrections

Abstract

We prove a new result about the mutual behavior of irrationality measure functions ψαj(t)ψ_{α_j}(t) for nn different real numbers αj, j=1,...nα_j,\, j =1,...n.

AI-generated audit

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 criteria and constructions for best approximations to nn real numbers are correct. Several denominator labels in the case analysis are local notation slips.

Theorems 1–3Correct

The simultaneous and dual best-approximation criteria have the stated form

Pages 2–13 · Theorems 1–3 and their proofs · arXiv:2108.08778v2

The convex-body minima translate bounded approximation constants into bounded ratios of successive best-approximation scales, and conversely. The recursive denominator construction separates each prescribed competitor and realizes the announced behavior.

Full paper, version 2 ↗
Case analysisMinor formal corrections · no status impact

Three denominator labels and one count require local correction

Pages 4–11 · proofs of Theorems 1–3 · arXiv:2108.08778v2

The second case of Theorem 1 must count 1+⌈(W1−1)/2⌉1+\lceil(W_1-1)/2\rceil admissible points. In Theorem 3, the prose swaps the two discontinuity denominators before returning to the correct common denominators. In the construction, the comparison in (32) is between the current common denominator and the two successive tt-indices; restoring those labels gives the displayed inequality.

Theorem 2Correct

The construction realizes the sharp order of the permutation bound

Pages 2 and 8–13 · Theorem 2 and Section 6 · arXiv:2108.08778v2

The prescribed interlacing sequence tjt_j determines continued fractions for the individual numbers and for each paired auxiliary number. The asymptotic denominator identities order all irrationality-measure functions into exactly the announced cyclic permutations. The special case k=3k=3 and the general index pattern satisfy the same inequalities, so the construction gives n=k(k+1)/2n=k(k+1)/2 numbers with only kk eventual permutations.

02Proofs1 reported findingCorrect

The geometric-minima arguments and realizing construction are complete after the denominator and counting labels are corrected.

Proofs of Theorems 1–3Correct and complete after the stated repairs

The corrected counts and denominator ordering close each case

Pages 3–13 · main proofs · arXiv:2108.08778v2

The corrected ceiling count restores the claimed lower bound without changing its constant order. The common-denominator comparisons in the construction strictly order the next approximation errors, which is the only property needed in the induction.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2108.08778v2
Authors listed
Vassily Manturov, Nikolay Moshchevitin
Audit date
August 20, 2026
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