[论文解读] 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 0

一句话总结

该论文证明在一类非线性非保守的地毯上，当满足坐标 OSC 条件时，糟糕可近似点具有全 Hausdorff 维数，并给出这些吸引子的一组 Hausdorff 维数公式。它还表明在合适的非线性地毯上，与 Bad_2 的交集达到全维数。

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.

研究动机与目标

• 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.

提出的方法

• 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.

实验结果

研究问题

• RQ1 Can Schmidt’s game methods yield full dimension of Bad_2 intersecting a non-linear non-conformal carpet?
• RQ2 What is the exact formula for the Hausdorff dimension of these non-linear carpets under coordinate OSC?
• RQ3 Does the modified lower dimension equal the Hausdorff dimension for these carpets, and under what conditions?
• RQ4 Under which structural conditions (columns/rows, hyperplane diffusivity) does X ∩ Bad_2 achieve dim_H(X)?
• RQ5 How do parabolic Cantor sets behave under similar methods?

主要发现

• 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.