Dimension estimates for badly approximable affine forms

TL;DR

This paper investigates the size and structure of sets of matrices and targets that are badly approximable in a Diophantine sense, providing upper bounds on their Hausdorff dimensions and conditions for full dimension.

Contribution

It establishes new upper bounds for Hausdorff dimensions of badly approximable sets and characterizes when these sets have full dimension for fixed matrices or targets.

Findings

Upper bounds for Hausdorff dimensions of badly approximable sets.

Conditions for full Hausdorff dimension of certain badly approximable sets.

Extension of methods to weighted approximation settings.

Abstract

For given $\epsilon>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $\epsilon$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq \epsilon,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $\epsilon$-badly approximable matrices for fixed target $b$ and the set of $\epsilon$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $\epsilon$-badly approximable targets for fixed $A$ has full Hausdorff dimension for some $\epsilon>0$. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the $A$-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.