Non-recurrence and divergent $(\mathbf{p}(\mathbf{n}),\mathbf{q}(\mathbf{n}))$-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 $p(n+1)-p(n),q(n+1)-q(n)\to +\mathrm{\infty }\mathrm{.}$ Then there exist automorphisms $S,T$ with simple singular spectrum and a set $C$ such that the sequence ${\sum }_{n=1}^N\mu (S^{p(n)}C\cap T^{q(n)}C)\mathrm{/}N$ 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.