Non-recurrence and divergent $\bf ({p(n)},{q(n)})$-averages for deterministic automorphisms

We answer the question of Frantzikinakis and Host about the convergence of ergodic $(n^2,n^3)$-averages and consider a more general case. Let sequences ${ p(n)},{ q(n)}$ satisfy the property $p(n+1)- p(n), \ q(n+1)- q(n)\ \to\ +\infty.$ Then there exist automorphisms $S,T$ with simple singular spectrum and a set $C$ such that the sequence $\sum_{n=1}^{N} \mu(S^{ p(n)}C\cap T^{ q(n)}C)/N$ diverges. We give also example of linear non-recurrence for a pair of mixing suspensions of zero entropy and with singular and Lebesgue parts in their spectra.