Examples on harmonic measure and normal numbers

Abstract

ABSTRACT. Suppose that F is a bounded set in Rm, m > 2, with positive capacity. Add to F a disjoint set E so that E U F is closed, and let D = Rm\ (ËUF).Under what conditions on the added set E do we have harmonic measure u(F, D) = 0? It turns out that besides the size of E near F, the location of E relative to F also plays an important role. Our example, based on normal numbers, stresses this fact.Suppose that F is a bounded set in Rm, m > 2, with positive capacity. Add to F a disjoint set E so that E U F is closed, and let D = Rm\(E U F). Under what conditions on the added set E do we have harmonic measure w(F, D) = 0? It turns out that besides the size of E near F, the location of E relative to F also plays an important role. Our example, based on normal numbers, stresses this fact.THEOREM. Let D be a bounded domain in Rm, m > 2, F be a subset of dD with Am~1(F) = 0, and E = Rm\(D U F). Suppose that F lies also on some quasi-smooth curve Y when m = 2, on some BMOi surface T when m > 3. And suppose that at each a G F, 0 < r < |, there is a closed set T Ç EC\B(a, r) so that (1) capacity ( However, when m = 2, (1) is more restrictive than (1'). The surface T and dD are in general distinct; no smoothness condition is imposed on dD. The problem is very different if dD is quasi-smooth or BMOi (see [4]). When m > 3, topological properties of D are less important in studying u(F,D):