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DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION

  • Victor Beresnevich (a1), Lawrence Lee (a2), Robert C. Vaughan (a3) and Sanju Velani (a4)
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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Mathematika
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