Fourier coefficients of a measure and absolute continuity

Question

A relative of a theorem of Peyriere (found in a 1997 paper of Klemes and Reinhold, "Rank One Transformations with Singular Spectral Type") says that if $\mu$ is a Borel probability measure on $S^1$ and there is $\{m_k\} \subset \mathbb{Z}$ such that setting $\hat{\mu}(m_k) = a_k$ yields $\hat{\mu}(m_k - m_j) = a_k\overline{a_j}$ and $\{a_k\} \notin \ell^2$, then $\mu$ is singular with respect to the Lebesgue measure on $S^1$. Are there any known conditions on the Fourier-Stieltjes coefficients of such a measure that would force it to be absolutely continuous with respect to the Lebesgue measure? Or even not singular?

Answer 1

The most obvious: $\hat{\mu} \in \ell^2$.