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

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Journal Mathematical Proceedings of the Cambridge Philosophical Society Accepted/In press - 23 Jan 2017 E-pub ahead of print - 5 Apr 2017 Published (current) - May 2018 3 164 47 413-459 5/04/17 English

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.

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© Cambridge Philosophical Society 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

• math.NT

• Programme Grant-New Frameworks in metric Number Theory

Project: Research project (funded)Research

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