Fourier-Stieltjes transforms with small supports

Abstract

Let $G$ be LCA group with dual $\Gamma$. Suppose $S\subseteq \Gamma$ is Borel set such that $S \cap (\gamma-S)$ has finite Haar measure for a dense set of $\gamma \epsilon \Gamma$ (or $S \cap (S-\gamma)$ does). If $\mu$ and $\nu$ are regular Borel measures whose Fourier-Stieltjes transforms vanish off $S$, then $|\mu|\ast|\nu|\epsilon|L^{1}(G)$ ($|\mu|$ denotes the total variation measure). This generalizes to non-metrizable groups a result of Glicksberg. Related results are given; the proofs are elementary.