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

Published online by Cambridge University Press:  05 April 2017

DAVID SIMMONS
DAVID SIMMONS*
Affiliation:
University of York, Department of Mathematics, Heslington, York YO10 5DD. e-mail: [email protected]
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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 generalising 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 ψ-approximable, given a dimension function f and a function ψ : (0, ∞) → (0, ∞). This complements earlier work by Dickinson and Velani ('97) who found the Hausdorff f-measure of the set of matrices which are ψ-approximable.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 413 - 459
Copyright
Copyright © Cambridge Philosophical Society 2017 

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