Salem-Schaeffer measures of dynamical system origin

A class of measure preserving actions of the groups $Z^d$ and $R^d$ is constructed possessing the following properties: the spectrum of the action is simple and for a dense set of functions $f$ the spectral measures $\sigma_f$ have an extremal rate of the Fourier coefficient decay: $\hat\sigma_f(n) = O(|n|^{-d/2+\varepsilon})$ for any $\varepsilon > 0$, where the exponent $-d/2$ is the minimal possible for singular mesures on the torus $T^d$.