Concentration functions in locally compact groups (Q1923275)

This article gives various theorems describing when the concentration functions of a probability measure on a locally compact Hausdorff group converge to 0. In the first section, this is proved assuming that the measure is irreducible, thereby giving a positive solution of the Hofmann-Mukherjea problem. In the second section, the same result is again proved for general groups, but with the assumption that the probability measure is adapted and almost aperiodic. The non-existence of strange locally compact groups is central to both of these results. In the third section, general issues connected with the concentration function problem and a possible representation theoretic approach are discussed.