  We study two problems concerning harmonic measure on "champagne subdomains"
of the unit disk. These domains are obtained by removing from the unit disk
little disks around sequences of points with a uniform distribution with
respect to the pseudohyperbolic metric of the unit disk. 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''.
