Density of the sums of shifts of a single function in the $L_2^0$ space on a compact Abelian group

Abstract

Let $G$ be a nontrivial compact Abelian group. The following result is proved: a real-valued function on $G$ such that the sums of shifts of it are dense in the $L_{2}$-norm in the corresponding real space of mean zero functions exists if and only if the group $G$ is connected and has an infinite countable character group. Bibliography: 13 titles.