Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem

The paper is primarily concerned with the asymptotic behavior as $N\to∞$ of averages of nonconventional arrays having the form ${N^{ - 1}}\sum\limits_{n = 1}^N {\prod\limits_{j = 1}^\ell {{T^{{P_j}(n,N)}}} } {f_j}$ where $f_j$'s are bounded measurable functions, $T$ is an invertible measure preserving transformation and $P_j$'s are polynomials of $n$ and $N$ taking on integer values on integers. It turns out that when $T$ is weakly mixing and $P_j(n, N) = p_jn+q_jN$ are linear or, more generally, have the form $P_j(n, N) = P_j(n)+Q_j(N)$ for some integer valued polynomials $P_j$ and $Q_j$ then the above averages converge in $L^2$ but for general polynomials $P_j$ of both $n$ and $N$ the $L^2$ convergence can be ensured even in the "conventional" case $\ell = 1$ only when $T$ is strongly mixing while for $\ell>1$ strong $2\ell$-mixing should be assumed. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemerédi's theorem saying that for any subset of integers $\Lambda $ with positive upper density there exists a subset ${\cal N}_\Lambda $ of positive integers having uniformly bounded gaps such that for $N∈{\cal N}_\Lambda $ and at least $\varepsilon N, \, \varepsilon >0$ of $n$'s all numbers $p_jn+q_jN, \, j = 1, ..., \ell, $ belong to $\Lambda $. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.