Title: Ergodic Theorems for Nonconventional Arrays and an Extension of the Szemeredi Theorem,
Abstract: The paper is primarily concerned with the asymptotic behavior as $N\to\infty$
of averages of nonconventional arrays having the form
$N^{-1}\sum_{n=1}^N\prod_{j=1}^\ell T^{P_j(n,N)}f_j$ 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 $P_j$ the $L^2$ convergence can be ensured
even in the case $\ell=1$ only when $T$ is strongly mixing. Studying also
weakly mixing and compact extensions and relying on Furstenberg's structure
theorem we derive an extension of Szemer\' edi's theorem saying that for any
subset of integers $\Lambda$ with positive upper density there exists a subset
$\mathcal N_\Lambda$ of positive integers having uniformly bounded gaps such
that for $N\in\mathcal N_\Lambda$ 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 Szemer\' edi theorem.