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

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JournalMathematical Proceedings of the Cambridge Philosophical Society
DateAccepted/In press - 23 Jan 2017
DateE-pub ahead of print - 5 Apr 2017
DatePublished (current) - May 2018
Issue number3
Volume164
Number of pages47
Pages (from-to)413-459
Early online date5/04/17
Original languageEnglish

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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