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DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION

Published online by Cambridge University Press:  29 November 2017

Victor Beresnevich ,
Lawrence Lee ,
Robert C. Vaughan  and
Sanju Velani
Victor Beresnevich
Affiliation:
University of York, Heslington, York YO10 5DD, U.K. email victor.beresnevich@york.ac.uk
Lawrence Lee
Affiliation:
University of York, Heslington, York YO10 5DD, U.K. email ldl503@york.ac.uk
Robert C. Vaughan
Affiliation:
Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802-6401, U.S.A. email rcv4@psu.edu
Sanju Velani
Affiliation:
University of York, Heslington, York YO10 5DD, U.K. email sanju.velani@york.ac.uk
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Abstract

Given $n\in \mathbb{N}$ and $\unicode[STIX]{x1D70F}>1/n$ , let ${\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ denote the classical set of $\unicode[STIX]{x1D70F}$ -approximable points in $\mathbb{R}^{n}$ , which consists of $\mathbf{x}\in \mathbb{R}^{n}$ that lie within distance $q^{-\unicode[STIX]{x1D70F}-1}$ from the lattice $(1/q)\mathbb{Z}^{n}$ for infinitely many $q\in \mathbb{N}$ . In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}>1/n$ almost all points on ${\mathcal{M}}$ are not $\unicode[STIX]{x1D70F}$ -approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set ${\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ . In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], which enables us to find a sharp lower bound for $\dim {\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ for any $C^{2}$ submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}$ satisfying $1/n\leqslant \unicode[STIX]{x1D70F}<1/m$ . Here $m$ is the codimension of ${\mathcal{M}}$ . We also show that the condition on $\unicode[STIX]{x1D70F}$ is best possible and extend the result to general approximating functions.

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Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 762 - 779
Copyright
Copyright © University College London 2017 

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