Metric Diophantine approximation on the middle-third Cantor set

TL;DR

This paper explores the Hausdorff dimension of numbers in the middle-third Cantor set that are well-approximated by rationals, proposing a conjecture and supporting it through probabilistic models and dimension estimates.

Contribution

It introduces a conjecture for the dimension of Diophantine approximation sets within the Cantor set and provides evidence using probabilistic models and dimension estimates.

Findings

The conjecture holds for a natural probabilistic model.

Dimension estimates support the proposed conjecture.

Results extend understanding of Diophantine approximation in fractal sets.

Abstract

Let $\mu\geq 2$ be a real number and let $\Mcal(\mu)$ denote the set of real numbers approximable at order at least $\mu$ by rational numbers. More than eighty years ago, Jarn\'i k and, independently, Besicovitch established that the Hausdorff dimension of $\Mcal(\mu)$ is equal to $2/\mu$. We investigate the size of the intersection of $\Mcal(\mu)$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\Mcal(\mu)$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The core of our study relies heavily on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points.