On Dirichlet spaces with zero sets containing all Blaschke sequences

Abstract We consider the family of infinite positive Borel measures $\mu $ in the unit disc, with finite degree of contact at the unit circle, and the set $E_{\mu }$ of points in the unit circle for which every neighborhood contains infinite mass of $\mu $ . Assuming that $E_{\mu }$ is a Carleson set, we show that all Blaschke sequences are zero sets for the corresponding Dirichlet space $D_{\mu }$ .