Growth conditions for thin sets in Vilenkin groups of bounded order

Abstract:

Let $G$ be a Vilenkin group of bounded order and ${H_n}$ a sequence of clopen subgroups of $G$ forming a base at the identity. If $E$ is a subset of $G$, let ${N_n}(E)$ denote the number of cosets of ${H_n}$ which intersect $E$. If \[ \underline {\lim } \frac {{{N_n}(E)}} {{\log [G:{H_n}]}} < \infty ,\] then $E$ is a U-set in the group $G$. It is also shown that for $G$ satisfying a growth condition and $\varphi (n) \to \infty$, there is an M-set, $E$, with \[ {N_n}(E) = O(\varphi (n) \log [G:{H_n}]).\]