Singular Schaeffer-Salem measures of dynamical system origin

We study a class of dynamical systems given by measure preserving actions of the group $Z^d$ or $R^d$ and generating a set of spectral measures with an extremal rate of the Fourier coefficient decay: $\Hat\sigma(n) = O(|n|^{-1/2+\epsilon})$ for any $\epsilon > 0$. Singular measures with this property are investigated in works due to Wiener and Wintner, Schaeffer, Salem, Ivashev-Musatov, Zygmund et al. Thus, the discovered effect provides a new construction of singular distributions of Schaeffer-Salem type on the torus $T^d$ and in the space $R^d$.