Metric Diophantine approximation on the middle-third Cantor set

Abstract

Let $\mu \ge 2$ be a real number and let $\mathcal{M}(\mu )$ denote the set of real numbers approximable at order at least $\mu$ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $\mathcal{M}(\mu )$ is equal to $2/\mu$. We investigate the size of the intersection of $\mathcal{M}(\mu )$ with Ahlfors regular compact subsets of the interval $[0,1]$. In particular, we propose a conjecture for the exact value of the dimension of $\mathcal{M}(\mu )$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We show in particular 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.