[論文レビュー] Badly approximable points on non-linear carpets

Roope Anttila, Jonathan M. Fraser+1 more|arXiv (Cornell University)|Mar 12, 2026

Mathematical Dynamics and Fractals被引用数 0

ひとこと要約

The paper proves that badly approximable points have full Hausdorff dimension on a class of non-linear non-conformal carpets under coordinate open set condition, and provides a Hausdorff dimension formula for these attractors. It also shows that the intersection with Bad_2 attains full dimension for suitable non-linear carpets.

ABSTRACT

The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urbański from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest.

研究の動機と目的

• Motivate and quantify how badly approximable points interact with non-linear, non-conformal fractal attractors.
• Extend Schmidt-game and lower-dimension methods to non-linear, non-conformal settings.
• Provide a Hausdorff-dimension formula for the non-linear carpets studied.
• Establish conditions under which Bad_2 intersects these carpets in full dimension.
• Relate the results to parabolic Cantor sets where applicable.

提案手法

• Define non-linear carpets as attractors of a planar IFS formed from coordinate self-conformal IFSs with a coordinate OSC.
• Use a symbolic Barański-carpet framework to approximate the non-linear carpets from inside.
• Prove a variational principle: dim_H(X) equals the supremum of dim_H(μ) over ergodic measures μ.
• Show dim_H(X) = dim_ML(X) = sup{dim_L(X′): X′ ⊂ X} under coordinate OSC.
• Utilize a Bernoulli-measure optimization on symbol spaces to derive dimensions, plus a bounded distortion lemma to control distortions.
• Apply a Schmidt-game–lower-dimension approach to obtain intersection results with Bad_d, particularly Bad_2.]
• research_questions:[

実験結果

リサーチクエスチョン

• RQ1 Schmidt’s game methods によって non-linear non-conformal カーペットと Bad_2 の交差が全次元を与えるか？
• RQ2 coordinate OSC 下でこれらの non-linear カーペットの Hausdorff 次元の正確な公式は何か？
• RQ3 これらのカーペットに対して改良された下次元は Hausdorff 次元と等しいか、どの条件下で？
• RQ4 構造条件（列/行、超平面拡散性）下で X ∩ Bad_2 が dim_H(X) を達成するのはどの条件か？
• RQ5 同様の方法でパラボリック Cantor 集合はどのように振る舞うか？

主な発見

• Dim_H(X) equals the supremum of dim_H(μ) over ergodic measures μ for non-linear carpets with coordinate OSC.
• Dim_H(X) equals dim_ML(X) and equals the supremum of dim_L(X′) over X′ ⊂ X for these carpets.
• If X satisfies coordinate OSC and has at least two maps in some column and some row, then dim_H(X ∩ Bad_2) = dim_H(X).
• The framework connects non-linear carpets to symbolic Barański carpets allowing dimension calculations via Bernoulli measures.
• The results provide a variational principle and constructive approach for numerics via subsystem approximations.
• The approach also yields insights for parabolic Cantor sets within this methodology.