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

For each integer $n>1$ a unitary operator of dynamical origin is found such that its $n$th tensor power has a singular spectrum, but the spectrum of the $(n+1)$st power is absolutely continuous. For any sequences $p(n)$ and $q(n)$, provided that $ p(n+1)- p(n) \to+\infty$ and $ q(n+1)- q(n)\to +\infty$, there exist a set $C$ and automorphisms $S$ and $T$ with simple singular spectra such that the sequence $ \sum_{n=1}^{N} \mu(S^{ p(n)}C\cap T^{ q(n)}C)/N$ is divergent. In the class of Poisson suspensions with zero entropy there exist mixing automorphisms $S$ and $T$ such that for some set $D$ of positive measure, $S^nD\cap T^nD=\varnothing$ for each $n>0$.

rank-one Sidon constructions, tensor products, spectrum, Poisson suspensions, joint dynamics, divergence of ergodic averages