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 \(H \le G\) 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 \(\Vert \mu \Vert \le 1\) and \(\widehat{\mu }(\widehat{G})\subset \{a_n: n \in \mathbb {N}\}\cup \{0\}\) , the measure \(\mu *\mu \) 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 the article ‘On the relationships between Fourier-Stieltjes coefficients and spectra of measures’ published in 2014.
