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 $P_j(n,N)=p_{j}n+q_{j}N$ 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 $\epsilon N,\epsilon >0$ of $n$'s all numbers $p_{j}n+q_{j}N,j=1,...,\ell$ 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.

28 pages

References in corpus (1)

• Weakly mixing sets and polynomial equations