  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.
