DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION

摘要

Given $nin mathbb{N}$ 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 $qin mathbb{N}$ . 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 t