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Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either areal-analytic manifold or the limit set of an iterated function system (forexample, $S$ 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 $S$. Theseapproximations correspond to a question asked by K. Mahler ('84) regarding theCantor 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 forevery $\mathbf x\in S\setminus\mathbb Q^d$, there exists $C > 0$ such thatinfinitely many vectors $\mathbf p/q\in \mathbb Q^d\setminus S$ satisfy$\|\mathbf x - \mathbf p/q\| < C/q^{(d + 1)/d}$. As this formula agrees withDirichlet's theorem in $\mathbb R^d$ up to a multiplicative constant, oneconcludes that the set of rational approximants to points in $S$ which lieoutside of $S$ is large. Furthermore, we deduce extrinsic analogues of theJarn\'ik--Schmidt and Khinchin theorems from known results.

Extrinsic Diophantine approximation on manifolds and fractals