# Ergodic Theorems for Nonconventional Arrays and an Extension of the Szemeredi Theorem

## TL;DR

This paper extends ergodic theorems and Szemerédi's theorem to nonconventional arrays involving polynomial iterates, establishing convergence results and combinatorial implications for subsets of integers with positive density.

## Contribution

It introduces new ergodic theorems for nonconventional arrays with polynomial iterates and extends Szemerédi's theorem to these settings, including multidimensional cases.

## Key findings

- Convergence of averages for weakly mixing transformations with linear polynomial iterates.
- Extension of Szemerédi's theorem ensuring structured subsets within dense integer sets.
- Results for multiple commuting transformations generalizing classical combinatorial theorems.

## 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.