Extrinsic Diophantine approximation on manifolds and fractals

TL;DR

This paper extends classical Diophantine approximation results to points on manifolds and fractals, showing that points outside these sets can be well-approximated by rationals, with implications for the structure of such approximations.

Contribution

It establishes an extrinsic analogue of Dirichlet's theorem for manifolds and fractals, demonstrating the abundance of rational approximations outside the set.

Findings

Infinitely many rational points outside the set approximate points in the set.

The approximation rate matches Dirichlet's theorem up to a constant.

Derived extrinsic versions of Jarník–Schmidt and Khinchin theorems.

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 $\|\mathbf x - \mathbf p/q\| < C/q^{(d + 1)/d}$. 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 Jarn\'ik--Schmidt and Khinchin theorems from known results.