Metric Diophantine Approximation on the Middle-third Cantor Set

Let $\Mcal(\mu)$ be the set of all real numbers that are approximable at a rate at least a given $\mu\geq 2$ by the rationals. More than eighty years ago, Jarní k and, independently, Besicovitch established that the Hausdorff dimension of $\Mcal(\mu)$ is equal to $2/\mu$. We consider the further question of 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. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. This study relies 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. This is a joint work with Yann Bugeaud (Strasbourg).