Published paper

The set of badly approximable vectors is strongly $C^1$ incompressible

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

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Publication metadata

Journal citation: Mathematical Proceedings of the Cambridge Philosophical Society 153(2), 319–339 (2012)
Publication date: 2012-05-23
DOI: 10.1017/S0305004112000242 ↗
arXiv counterpart: arXiv:1106.1621v1

Abstract

The paper introduces the hyperplane absolute winning game, proves its countable-intersection and invariance properties, and shows that hyperplane absolute winning sets are winning for Schmidt's original game.

Role in dependence graphs

Proof-critical source

On Some Properties of Irrational Subspaces

This paper is included only for the following marked statement:

HAW permanence properties · §2HAW sets are stable under countable intersections and imply winning for Schmidt's game.

Proof-critical source

Metric theory with a fixed matrix

This paper is included only for the following marked statement:

Proposition 2.3(a)–(b) · printed p. 323HAW implies Schmidt winning/full Hausdorff dimension, and countable intersections of HAW sets are HAW. The focal paper uses the intersection property in Lemma 7.8 and recalls the dimension implication.

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Version of record · Mathematical Proceedings of the Cambridge Philosophical Society 153 (2012), no. 2, 319–339

Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, Barak Weiss. The set of badly approximable vectors is strongly $C^1$ incompressible. Mathematical Proceedings of the Cambridge Philosophical Society 153 (2012), no. 2, 319–339.

Open audited source ↗

Generated August 23, 2026

01Statements3 reported findingsCorrect

The hyperplane-absolute-winning result for badly approximable vectors, its $C^1$ invariance, the strong incompressibility consequence, the fractal-intersection results, and the nondense-orbit example are correct. Three uniquely determined local typographical corrections do not alter any statement.

Theorems 2.4 and 2.5Correct

Hyperplane absolute winning is $C^1$ invariant and contains the badly approximable vectors

Journal pages 323–327 · Theorems 2.4–2.5 and their proofs · version of record

For Theorem 2.4, uniform control of $Df$ and $D(f^{-1})$ on the first sufficiently small compact game ball permits a subsequence of Bob's moves to be mapped to a legal game with parameter $\beta'=\beta^n$ . The affine linearization error is made smaller than the deletion margin, so pulling a deleted $k$ -plane neighborhood back gives a legal deletion and transfers Alice's winning strategy. For Theorem 2.5, the simplex lemma confines every denominator block to one affine hyperplane, and deleting its neighborhood yields the uniform bound $\lVert x-p/q\rVert\geq c q^{-(1+1/d)}$ . The printed move index $A_{j_k+1}$ must be $A_{j_k}$ ; with this mechanical correction the deletion is made immediately after $B_{j_k}$ , has admissible width $\beta^{k+1}\rho<\beta\rho_{j_k}$ , and forces $B_{j_k+1}$ to avoid the dangerous rational points.

Cambridge version of record ↗

Theorems 1.1 and 1.2Correct

The strong incompressibility and diffuse-fractal conclusions follow

Journal pages 320–321 and 330–334 · Theorems 1.1, 1.2, 4.10 and Corollary 5.4 · version of record

Hyperplane absolute winning is preserved by countable intersections and by inverse images under nonsingular $C^1$ maps. Proposition 4.9 transfers the game to every hyperplane-diffuse closed set, and the winning-on- $K$ dimension lemma gives positive dimension. When $K$ supports an Ahlfors-regular absolutely decaying measure, its Hausdorff dimension is recovered in full. Applying these facts to $\mathrm{BA}_d$ proves both advertised incompressibility conclusions with the stated quantifiers over the maps and open set.

Cambridge version of record ↗

Theorem 2.6Correct

Avoidance sets for semisimple toral endomorphisms are HAW

Journal pages 324 and 327–330 · Theorem 2.6 and proof · version of record

If the spectral radius is one, Kronecker's theorem and semisimplicity make a common power of the integral rational matrix equal to the identity, reducing the target to avoidance of a countable set. If the spectral radius exceeds one, the proof separates the maximal-modulus invariant subspace, compares different inverse images of the target lattice, and deletes the affine hyperplane parallel to the complementary invariant subspace at each selected scale. The separation estimates ensure that every surviving orbit stays a fixed distance from the target point.

Cambridge version of record ↗

02Proofs4 reported findingsCorrect

The material proofs are correct and complete after three mechanical typographical corrections. The pullback strategy, simplex-lemma argument, diffuse-set restriction, and dimension deductions preserve all parameter ranges and quantifiers.

Proof of Theorem 2.4Correct and complete

The game pullback controls nonlinear distortion at every selected scale

Journal pages 324–327 · proof of Theorem 2.4 · version of record

The choice $C=2C_1C_2$ and $n$ with $C(1+\beta)\beta^{n-2}<1$ makes the pulled-back deletion thinner than $\beta\rho_i$ . The chosen subsequence has radius ratio in $[\beta^n,\beta^{n-1})$ , which supplies a legal $\beta^n$ -game and leaves the required separation from the preceding deleted plane. Continuity of the inverse derivative makes the affine approximation error uniform on all later balls. Thus every outcome of the original game maps into an outcome of the transferred winning strategy.

Cambridge version of record ↗

Proof of Theorem 2.5Typo

The deleted-plane move has an off-by-one printed index

Journal page 327 · proof of Theorem 2.5, display defining Alice's move · version of record

The display prints $A_{j_k+1}=\mathcal L_k^{(\beta^{k+1}\rho)}$ , although Alice must answer Bob's already chosen ball $B_{j_k}$ . The unique correction is $A_{j_k}=\mathcal L_k^{(\beta^{k+1}\rho)}$ . It is legal because $\rho_{j_k}>\beta^k\rho$ , and then the next ball $B_{j_k+1}$ avoids the neighborhood exactly as the following sentence requires. No theorem or later application changes.

Cambridge version of record ↗

Proposition 5.1Typo

The displayed strict comparison should be an equality

Journal page 334 · proof of Proposition 5.1 · version of record

After choosing $\beta=(1/C)^{1/\gamma}$ , the proof says $C\beta^\gamma<1$ ; the left side is exactly $1$ . Replacing $<$ by $=$ is the unique correction. The defining decay estimate itself is strict, so it still gives $\mu(B(x,\rho)\cap\mathcal L^{(\beta\rho)})<\mu(B(x,\rho))$ , leaving a support point outside the hyperplane neighborhood and proving diffuseness without any additional argument.

Cambridge version of record ↗

Lemma 5.3Typo

The open-set variable is misnamed once

Journal page 334 · statement of Lemma 5.3 · version of record

The lemma quantifies an open set $U$ but ends its hypothesis with $V\cap K\neq\varnothing$ . The unique intended correction is $U\cap K\neq\varnothing$ , as in the preceding and following statements. The corrected hypothesis is exactly the one used to deduce Corollary 5.4.

The set of badly approximable vectors is strongly C1C^1C1 incompressible

Publication metadata

Abstract

Role in dependence graphs

On Some Properties of Irrational Subspaces

Metric theory with a fixed matrix

Audit summary

Version of record · Mathematical Proceedings of the Cambridge Philosophical Society 153 (2012), no. 2, 319–339

Hyperplane absolute winning is C1C^1C1 invariant and contains the badly approximable vectors

The strong incompressibility and diffuse-fractal conclusions follow

Avoidance sets for semisimple toral endomorphisms are HAW

The game pullback controls nonlinear distortion at every selected scale

The deleted-plane move has an off-by-one printed index

The displayed strict comparison should be an equality

The open-set variable is misnamed once

The set of badly approximable vectors is strongly $C^1$ incompressible

Hyperplane absolute winning is $C^1$ invariant and contains the badly approximable vectors