Champagne subdomains with unavoidable bubbles

A champagne subdomain of a connected open set U≠0̸ in Rd, d≥2, is obtained by omitting pairwise disjoint closed balls B¯(x,rx), x∈X, the bubbles, where X is an infinite, locally finite set in U. The union A of these balls may be unavoidable, that is, Brownian motion, starting in U∖A and killed when leaving  U, may hit  A almost surely or, equivalently, A may have harmonic measure 1 for U∖A.Recent publications by Gardiner and Ghergu (d≥3) and by Pres (d=2) give rather sharp answers to the question of how small such a set A may be, when U  is the unit ball.In this paper, using a totally different approach, optimal results are obtained, which hold also for arbitrary connected open sets  U.