Champagne subregions of the disk whose bubbles carry harmonic measure (Q697320)

Let \(G\) be a region in the unit dik \(D\) such that the boundary of the unbounded component of \(\mathbb{C}\setminus\overline G\) is the unit circle \(\partial\mathbb{D}\). It is proved that for any \(\varepsilon> 0\) there exists a sequence \(\{\Delta_n\}\) of pairwise disjoint closed disks in \(G\) whose set of accumulation points coincides with \(\partial\mathbb{D}\), the sum of the radii is less than \(\varepsilon\), and the harmonic measure of \(\mathbb{D}\setminus\bigcup_n \Delta_n\) does not charge \(\partial\mathbb{D}\).