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

David Simmons

doi:10.1017/S0305004117000214

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

David Simmons

Mathematics

Research output: Contribution to journal › Article › peer-review

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
Pages (from-to)	413-459
Number of pages	47
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	164
Issue number	3
Early online date	5 Apr 2017
DOIs	https://doi.org/10.1017/S0305004117000214
Publication status	Published - May 2018

Bibliographical note

© 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

Keywords

math.NT

Access to Document

10.1017/S0305004117000214

1509.03885
© 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 (embargo period: 6 months)
Accepted author manuscript, 427 KB

https://arxiv.org/abs/1509.03885

Programme Grant-New Frameworks in metric Number Theory
Velani, S. & Beresnevich, V.
EPSRC
1/06/12 → 30/11/18
Project: Research project (funded) › Research

Cite this

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

Abstract

Bibliographical note

Keywords

Access to Document

Projects

Programme Grant-New Frameworks in metric Number Theory

Cite this