Wiener-Pitt sets for compact Abelian groups

Suppose that $G$ is a compact Hausdorff Abelian group. We say $\mu \in M(G)$ is strongly continuous if $|\mu|(x+H)=0$ for any $x \in G$ and any $H \leq G$ that is closed and of infinite index. We prove that for any sufficiently rapidly decreasing sequence $(a_{n})_{n=1}^{\infty}\in c_{0}(\mathbb{N})$, for every strongly continuous $\mu\in M(G)$ with $\|\mu\| \leq 1$ and $\widehat{\mu}(\widehat{G})\subset \{a_n: n \in \mathbb{N}\}\cup\{0\}$, the measure $\mu\ast\mu$ is absolutely continuous with respect to Haar measure on $G$. This implies that $\mu$ does not exhibit the so-called Wiener-Pitt phenomenon. The paper is a continuation of investigations started in \cite{ow}.