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

Singular Schaeffer-Salem measures of dynamical system origin