Convolutions of continuous measures and sums of an independent set

Abstract:

Let $E$ be a compact independent subset of an l.c.a. group $G;{\mu _1}, \cdots ,{\mu _{n + 1}}$ continuous regular bounded Borel measures on $G$; and ${k_1}, \cdots ,{k_n}$ integers. Let ${k_i} \times E = \{ {k_i}x|x \in E\}$. We prove (1) ${\mu _1} \ast \cdots \ast {\mu _{n + 1}}({k_1} \times E + \cdots + {k_n} \times E) = 0$ (the proof is a combinatorial argument). As a corollary of (1) we obtain (2) if $H$ is any closed nondiscrete subgroup of $G$, then the intersection of $H$ with the group generated by $E$ has zero $H$-Haar measure.