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arXiv:1106.1621v1

The set of badly approximable vectors is strongly C1C^1 incompressible

Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, Barak Weiss

math.NTmath.DS
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Paper metadata

Original submission
June 8, 2011
Claimed MSC
arXiv categories
math.NT, math.DS
Comments
18 pages

Abstract

We prove that the countable intersection of C1C^1-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in Rd\mathbb{R}^d, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's (α,β)(α, β)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.

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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 19, 2026
01Statements2 reported findingsCorrect

The hyperplane-absolute-winning theorem for badly approximable vectors and the strong differentiable incompressibility conclusions on diffuse fractals are correct.

Theorems 2.4 and 2.5Correct

Badly approximable vectors are hyperplane absolute winning

Pages 5–7 and 10–13 · Theorems 2.4–2.5 · arXiv:1106.1621v1

The simplex lemma places rational approximants of each denominator scale near one affine hyperplane, and deleting that hyperplane neighborhood forces the outcome to satisfy a uniform Diophantine lower bound. The game is invariant under the stated differentiable changes of variables.

Full paper, version 1 ↗
Theorems 1.1 and 1.2Correct

Strong incompressibility and fractal dimension follow

Pages 3–4 and 13–18 · main consequences · arXiv:1106.1621v1

Absolute decay plus Ahlfors regularity implies hyperplane diffuseness and supplies the dimension lower bound for winning sets. Closure under countable intersections and invariance under nonsingular differentiable maps give exactly the announced strong incompressibility.

02Proofs2 reported findingsCorrect

The simplex-lemma strategy and the diffuse-set dimension arguments are correct and complete.

Proof of Theorem 2.5Correct and complete

One deleted hyperplane controls each denominator block

Pages 10–13 · proof of Theorem 2.5 · arXiv:1106.1621v1

The denominator ranges are synchronized with game radii, the simplex lemma confines all dangerous rationals in the current ball, and diffuseness leaves a valid response. The limiting point consequently avoids every forbidden rational neighborhood.

Sections 4–5Correct and complete

The restriction and incompressibility machinery closes

Pages 13–18 · diffuse restriction and applications · arXiv:1106.1621v1

The induced game on a diffuse set remains playable at all small scales, and the mass-distribution estimate gives full dimension relative to that set. Local bi-Lipschitz control of the differentiable maps completes the countable intersection argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1106.1621v1
Authors listed
Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, Barak Weiss
Audit date
August 19, 2026
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