# A Hausdorff measure version of the Jarník-Schmidt theorem in Diophantine approximation

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## 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.
Original language English 413-459 47 Mathematical Proceedings of the Cambridge Philosophical Society 164 3 5 Apr 2017 https://doi.org/10.1017/S0305004117000214 Published - May 2018

### Bibliographical note

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## Keywords

• math.NT
• ### Programme Grant-New Frameworks in metric Number Theory

EPSRC

1/06/1230/11/18

Project: Research project (funded)Research