Let $\mu \ge 2$ be a real number and let $ℳ\left(\mu \right)$ 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 $ℳ\left(\mu \right)$ is equal to $2/\mu$. We investigate the size of the intersection of $ℳ\left(\mu \right)$ with Ahlfors regular compact subsets of the interval $[0,1]$. In particular, we propose a conjecture for the exact value of the dimension of $ℳ\left(\mu \right)$ intersected with the middle-third Cantor set and give several results...

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan’s classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd positive integers (Nesterenko, 2011).