On the Steinhaus property in topological groups

Let G be a locally compact Abelian group and mu a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is mu-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number In such that to n(0) <= m <= \G\ there is a subgroup of G of index m that is It-thick and everywhere of second category. These results extend theorems by Muthuvel and Erdos-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits 2(\G\)-many mu-nonmeasurable dense subgroups. (C) 2005 Elsevier B.V. All rights reserved.