A Hausdorff measure version of the Jarn\'ik--Schmidt theorem in Diophantine approximation

TL;DR

This paper establishes precise bounds on the Hausdorff dimensions of badly approximable matrices, extending classical results to higher dimensions and computing related Hausdorff measures, advancing the understanding of Diophantine approximation sets.

Contribution

It provides sharp asymptotic bounds on Hausdorff dimensions of certain Diophantine sets and computes their Hausdorff measures, generalizing previous results to higher dimensions.

Findings

Sharp bounds on Hausdorff dimensions of badly approximable matrices.

Calculation of Hausdorff $f$-measure for non-$\psi$-approximable matrices.

Extension of classical theorems to higher-dimensional settings.

Abstract

We solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalizing to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorff $f$-measure of the set of matrices which are not $\psi$-approximable, given a dimension function $f$ and a function $\psi:(0,\infty)\to (0,\infty)$. This complements earlier work by Dickinson and Velani ('97) who found the Hausdorff $f$-measure of the set of matrices which are $\psi$-approximable.