Harmonic measure and uniform densities

Abstract

We study two problems concerning harmonic measure on certain 'champagne subdomains' of the unit disk $\D$. The domains that we consider are obtained by removing from $\D$ little disks around sequences of points with a uniform distribution with respect to the pseudohyperbolic metric of $\D$. We find (I) a necessary and sufficient condition on the decay of the radii of the little disks for the exterior boundary to have positive harmonic measure, and (II) describe sampling and interpolating sequences for Bergman spaces in terms of the harmonic measure on such 'champagne subdomains'.