On the Steinhaus property in topological groups

Let G be a locally compact Abelian group and μ 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 μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ 0 ⩽ m ⩽ | G | there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erdős–Marcus, respectively. (b) also implies a recent theorem by Comfort–Raczkowski–Trigos stating that every nondiscrete compact Abelian group G admits 2 | G | -many μ-nonmeasurable dense subgroups.