Hausdorff Measures, Dyadic Approximation and Dobinski Set

We study the size of the Dobinski set, a subset of the unit interval defined in terms of dyadic approximation. This can be done by using two distinct approaches: first, we give an analogous of Jarnik’s Theorem in Diophantine approximation that fits our dyadic setting, and we use such a result ( together with some reference to quasi-independence) to prove that the Dobinski set has logarithmic Hausdorff dimension equal to 1, with associated logarithmic Hausdorff measure equal to infinity. We will also see how the same result can be proven by using technical estimates on the Hausdorff dimension for willow Cantor sets constructed inside the Dobinski set. This is a joint work with José Luís Fernández and Maria Jose González.