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 $\mathrm{\mid }\mu \mathrm{\mid }(x+H)=0$ for any $x\in G$ and any $H\le G$ that is closed and of infinite index. We prove that for any sufficiently rapidly decreasing sequence $(a_n{)}_{n=1}^{\mathrm{\infty }}\in c_0(\mathbb{N})$, for every strongly continuous $\mu \in M(G)$ with $\mathrm{\parallel }\mu \mathrm{\parallel }\le 1$ 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}.