Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions

TL;DR

This paper extends Hausdorff measure results for sets of matrices that are not improvable in Dirichlet's theorem, providing a more general measure-theoretic framework in higher dimensions.

Contribution

It generalizes previous Hausdorff measure results to a broader class of functions for Dirichlet non-improvable matrices in higher dimensions.

Findings

Established a generalized Hausdorff $f$-measure criterion

Extended Lebesgue measure results to Hausdorff measure

Provided new measure-theoretic insights into Dirichlet non-improvability

Abstract

Let $\psi:\mathbb R_{+}\to \mathbb R_{+}$ be a nonincreasing function. A pair $(A,\mathbf b),$ where $A$ is a real $m\times n$ matrix and $\mathbf b\in\mathbb R^{m},$ is said to be $\psi$-Dirichlet improvable, if the system $$\|A\mathbf q +\mathbf b-\mathbf p\|^m<\psi(T), \quad \|\mathbf q\|^n<T$$ is solvable in $\mathbf p\in\mathbb Z^{m},$ $\mathbf q\in\mathbb Z^{n}$ for all sufficiently large $T$ where $\|\cdot\|$ denotes the supremum norm. For $\psi$-Dirichlet non-improvable sets, Kleinbock--Wadleigh (2019) proved the Lebesgue measure criterion whereas Kim--Kim (2021) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff $f$-measure version of Kim--Kim (2021) results for $\psi$-Dirichlet non-improvable sets.