Extrinsic Diophantine approximation on manifolds and fractals

Abstract

Fix $d\in \mathbb{N}$, and let $S\subseteq {\mathbb{R}}^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine approximation to a point $\mathbf{x}\in S$ is a rational point $\mathbf{p}/q$ close to $\mathbf{x}$ which lies $outside$ of $S$. These approximations correspond to a question asked by K. Mahler ('84) regarding the Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem. Specifically, we prove that if $S$ does not contain a line segment, then for every $\mathbf{x}\in S\setminus {\mathbb{Q}}^d$, there exists $C>0$ such that infinitely many vectors $\mathbf{p}/q\in {\mathbb{Q}}^d\setminus S$ satisfy $\Vert \mathbf{x}-\mathbf{p}/q\Vert <C/q^{(d+1)/d}$. As this formula agrees with Dirichlet's theorem in ${\mathbb{R}}^d$ up to a multiplicative constant, one concludes that the set of rational approximants to points in $S$ which lie outside of $S$ is large. Furthermore, we deduce extrinsic analogues of the Jarník--Schmidt and Khinchin theorems from known results.