Hausdorff measure of sets of Dirichlet non-improvable affine forms

TL;DR

This paper investigates the Hausdorff measure of sets of affine forms that cannot be improved in Dirichlet approximation, extending previous Lebesgue measure results and computing Hausdorff dimensions for certain Diophantine exponents.

Contribution

It establishes a Hausdorff measure criterion for Dirichlet non-improvable sets and extends the analysis to fixed translation vectors, providing new dimension results.

Findings

Hausdorff measure criterion for non-improvable sets

Extension to fixed translation vector case

Calculation of Hausdorff dimension for Diophantine exponents

Abstract

For a decreasing real valued function $\psi$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $\psi$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < \psi(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $\psi$-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the $\psi$-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that $\mathbf{b}$ is fixed. As an application, we compute the Hausdorff dimension of the set of pairs $(A,\mathbf{b})$ with uniform Diophantine exponents $\widehat{w}(A,\mathbf{b})\leq w$.