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

Abstract

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.

Locations

• Mathematical Proceedings of the Cambridge Philosophical Society - View
• arXiv (Cornell University) - View - PDF
• White Rose Research Online (University of Leeds) - View - PDF