On the Steinhaus property in topological groups

Abstract

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.

Finite Partitions of Topological Groups into Congruent Thick Subsets

“…We have recently extended Erdős and Marcus' result to a rather general class of locally compact Abelian groups [14] equipped with a Haar measure µ.…”

“…A subset A of X is called µ-thick if the only Borel sets contained in X \ A are locally µ-null sets. 1 As observed in [14], if X is decomposed into k disjoint µ-thick subsets A 1 , . .…”

“…We also study finite decompositions of a Baire topological group into congruent sets that are "thick" in a topological sense, i.e., everywhere of second category. To treat the two notions of µ-thick sets and everywhere of second category sets simultaneously, in the spirit of [14] we first examine "thickness" with respect to a generic ideal I (see Definition 2.1). Both special cases we are mainly interested in are then obtained by taking for I the ideal of locally µ-null sets or the ideal of first category sets.…”