Hausdorff measures, dyadic approximations, and the Dobiński set

Abstract

The Dobiński set $\mathcal{D}$ is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of $\mathcal{D}$ by means of the mass transference principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in $\mathcal{D}$.