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arXiv:1311.5474v5

Dimension estimates for sets of uniformly badly approximable systems of linear forms

Ryan Broderick, Dmitry Kleinbock

math.NTmath.DS
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Paper metadata

Original submission
November 21, 2013
Latest revision
October 9, 2015 · version 5
Claimed MSC
arXiv categories
math.NT, math.DS
Comments
15 pages

Abstract

The set of badly approximable m×nm \times n matrices is known to have Hausdorff dimension mnmn . Each such matrix comes with its own approximation constant cc, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed cc. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions.

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Audit summary

Audited against arXiv v5

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 qualitative upper and quantitative lower codimension bounds are correct. The explicit exponent in Theorem 2.10 is also correct: although the printed sketch gives the wrong conversion exponent, the exact systems in the cited HAW proof yield a fully verified repair with the stated value of pp.

Theorems 1.2 and 1.3Correct

Qualitative upper and quantitative lower codimension bounds

Pages 2–4 · Theorems 1.2–1.3 · arXiv:1311.5474v5

The hyperplane-absolute-winning strategy supplies some exponent p(m,n)>0p(m,n)>0 for the upper codimension bound, and the homogeneous-space covering argument in Section 3 supplies the stated lower bound. Neither conclusion requires the particular value of pp printed in Theorem 2.10.

Full paper, version 5 ↗
Theorem 2.10Correct

The explicit exponent follows after a verified repair of the sketch

Pages 7–8 · Theorem 2.10 and its proof sketch · arXiv:1311.5474v5

Write L=m+nL=m+n and ℓ=max⁡{(4n+1)/m,(4m+1)/n}\ell=\max\{(4n+1)/m,(4m+1)/n\}. The cited HAW construction uses δ=R−nL2\delta=R^{-nL^2} and shows that absence of integer solutions to its first system implies A∈Badm,n(c)A\in Bad_{m,n}(c) with c=δLR−mL=R−[mL+nL3]c=\delta^L R^{-mL}=R^{-[mL+nL^3]}. The quantitative game estimate in the sketch gives R=Kβ−ℓR=K\beta^{-\ell}, hence c=K′βℓ[mL+nL3]=K′βpc=K'\beta^{\ell[mL+nL^3]}=K'\beta^p. Corollary 2.4 then gives codim⁡(Badm,n(c))=O(c1/p/log⁡(1/c))\operatorname{codim}(Bad_{m,n}(c))=O(c^{1/p}/\log(1/c)), exactly as asserted. This repairs the erroneous displayed conversion in the sketch without changing the theorem.

Cited HAW proof, version 1 ↗
02Proofs3 reported findingsContains incorrect or incomplete proofs

The displayed parameter conversion in the sketch of Theorem 2.10 is incorrect as written, but the cited systems provide a verified replacement that gives the asserted exponent. The other central dimension estimates are proved correctly. The limit direction in the reformulation of Theorem 2.10 is also a typo.

Sketch of Theorem 2.10Incorrect as written · verified repair

The displayed beta-to-approximation conversion has the wrong exponent

Pages 7–8 · sketch of proof of Theorem 2.10 · arXiv:1311.5474v5

The sketch prints c=K′β−ℓ(n(m+n)−m(m+n)3)c=K'\beta^{-\ell(n(m+n)-m(m+n)^3)}, which does not yield the stated pp. Repair classification: Verified repair. In the cited construction, with L=m+nL=m+n and δ=R−nL2\delta=R^{-nL^2}, the first no-solution system gives c=δLR−mL=R−[mL+nL3]c=\delta^L R^{-mL}=R^{-[mL+nL^3]}. Substitution of the sketch's R=Kβ−ℓR=K\beta^{-\ell} gives c=K′βℓ[mL+nL3]=K′βpc=K'\beta^{\ell[mL+nL^3]}=K'\beta^p, and Corollary 2.4 supplies the claimed bound.

Cited HAW proof, version 1 ↗
Section 3Correct and complete

The lower codimension estimate is independently complete

Section 3 · proof of Theorem 1.3 · arXiv:1311.5474v5

Quantitative Dani correspondence identifies Badm,n(c)Bad_{m,n}(c) with trajectories avoiding a cusp neighborhood of radius c1/(m+n)c^{1/(m+n)}. Mixing gives a uniform positive proportion entering the cusp at separated times, and the resulting covering count gives the stated c/log⁡(1/c)c/\log(1/c) codimension lower bound.

Reformulation after Theorem 2.10Typo

The small-constant limit is printed in the wrong direction

Page 7 · sentence following equation (2.1) · arXiv:1311.5474v5

The sentence ends ‘as c→∞c\to\infty,’ but Badm,n(c)Bad_{m,n}(c) and every bound in the paper are being considered for 0<c<10<c<1. Replace it by ‘as c→0c\to0.’ This is uniquely determined by the displayed log⁡(1/c)\log(1/c) and does not affect the separate exponent gap.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1311.5474v5
Authors listed
Ryan Broderick, Dmitry Kleinbock
Audit date
August 19, 2026
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