MathAudit

Paper metadata

Original submission
August 7, 2015
Latest revision
August 10, 2015 · version 2
Claimed MSC
arXiv categories
math.NT

Abstract

Let KK be a number field, let SS be the set of all normalized, non-conjugate Archimedean valuations of KK, and let KS=∏v∈SKvK_{S} = \prod_{v \in S} K_v be the Minkowski space associated with KK. We strengthen recent results of Esdahl--Kristensen and Einsiedler--Ghosh--Lytle by showing that the set of badly approximable elements of KSK_S is H\mathcal{H}-absolute winning for a certain family of subspaces of KSK_{S}.

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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 19, 2026
01Statements2 reported findingsCorrect

The hyperplane-family absolute-winning theorem for badly approximable elements of a number field's Minkowski space and its curve and fractal corollaries are correct. One free valuation index is a harmless typo.

Main TheoremCorrect

Badly approximable elements are absolute winning for the stated family

Page 3 and Section 5 (pages 14--18) · main theorem and proof · arXiv:1508.01770v2

The height version of Dani's correspondence converts bad approximation into a uniform lower bound along the diagonal flow. At each game scale, the refined simplex lemma shows that every potentially short primitive vector comes from one field fraction. Deleting the finitely many coordinate subspaces through that fraction makes the logarithmic height derivative nonnegative for every remaining short vector. The bounded time gaps then give a uniform positive height lower bound.

Corollaries 1.4--1.6Correct

The curve and Ahlfors-regular consequences follow from diffuseness

Pages 3--4 and Sections 4--5 · inheritance corollaries · arXiv:1508.01770v2

A curve satisfying the weighted nonvanishing condition is diffuse with respect to the forbidden coordinate subspaces outside its countable exceptional set, so inheritance converts the ambient winning strategy into ordinary absolute winning on the parameter interval. In the imaginary quadratic case the forbidden subspaces are points, and Ahlfors regular support is point-diffuse, giving the stated winning conclusion.

02Proofs2 reported findingsCorrect

The height estimates, uniqueness lemma, derivative criterion, and game strategy are correct and complete. The only identified mathematical notation defect has a unique local repair.

Section 5Correct and complete

The one-fraction reduction and game strategy close the argument

Pages 14--18 · Lemmas 5.1--5.6 and proof of the main theorem · arXiv:1508.01770v2

The determinant norm is a nonzero integer unless two fractions coincide, yielding the uniqueness threshold. Changing the base point within a game ball changes the height by at most the displayed factor. The derivative formula weights real places once and complex places twice, exactly matching the definition of the deletable subspace family. Lemma 6.4 then propagates discrete-time lower bounds to the full orbit.

Weak Dirichlet theoremTypo

The denominator norm contains a free valuation index

Page 5 · displayed inequality in the Weak Dirichlet Theorem · arXiv:1508.01770v2

The display ends with C∥ιv(q)∥−1C\|\iota_v(q)\|^{-1} although vv is not quantified there and the norm has been defined on the full Minkowski embedding. The strong theorem immediately above and the proof require C∥ιS(q)∥−1C\|\iota_S(q)\|^{-1}. Replacing vv by SS is uniquely determined and does not affect any later use.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1508.01770v2
Authors listed
Dmitry Kleinbock, Tue Ly
Audit date
August 19, 2026
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