Spectra and joint dynamics of Poisson suspensions for rank-one automorphisms

For every natural $n>1$, there is an operator $T$ of dynamical origin such that its tensor power $T^{\otimes n}$ has singular spectrum, and $T^{\otimes (n+1)}$ has absolutely continuous one. For a set $D$ of positive measure there are mixing zero entropy automorphisms $S,T$ such that $S^{n}D\cap T^{n}D=\mathrm{\varnothing }$ for all $n>0$. The following answers, in particular, Frantzikinakis-Host’s question. If $p(n+1)-p(n),q(n+1)-q(n)\to +\mathrm{\infty }$, then there is a divergent sequence ${\sum }_{n=1}^N\mu (S^{p(n)}C\cap T^{q(n)}C)\mathrm{/}N$ for some set $C$ and automorphisms $S,T$ of simple singular spectrum.