For each integer n>1 a unitary operator of dynamical origin is found such that its nth 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) such that p (n+1) - p (n) + and q (n+1) - q (n) + there exist a set C and automorphisms S and T with simple singular spectra such that the sequence ₍=₁^N (S^ p (n) C 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, SⁿD TⁿD= for all n>0. Bibliography: 23 titles.
V. V. Ryzhikov (Thu,) studied this question.