Polynomial rigidity and spectrum of Sidon automorphisms

We present spectrally disjoint Sidon automorphisms whose tensor squares are isomorphic to a planar shift. Spectra of such automorphisms do not possess the group property. To check the singularity of spectrum, we use polynomial rigidity of operators associated with Kolmogorov linear determinism. In the class of mixing Gaussian and Poisson suspensions we realize new sets of spectral multiplicities.