On the Foiaş and Strătilă theorem (Q6580006)

\textit{C. Foiaş} and \textit{S. Stratila} [C. R. Acad. Sci., Paris, Sér. A 267, 166--168 (1968; Zbl 0218.60040)] showed that if the spectral measure of a non-zero square-integrable complex function \(f\) on an ergodic measure-preserving system \((X,T)\) is continuous and supported on a Kronecker set then the process \((f\circ T^n)\) is Gaussian. In particular, the dynamical system generated by this process is determined up to meaurable isomorphism by the spectral measure. This is the only substantial setting of spectral determination in ergodic theory outside the discrete spectrum setting. These special `Gaussian-Kronecker' systems enjoy many special properties, and it is possible to completely describe their factors and their self-joinings by work of \textit{J.-P. Thouvenot} [Teubner-Texte Math. 94, 195--198 (1987; Zbl 0653.28014)]. Here the Foiaş and Strătilă theorem is extended to spectral measures that are continuous and concentrated on an independent Helson set and to ergodic actions of locally compact second countable abelian groups.