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- W2949492065 abstract "The paper is primarily concerned with the asymptotic behavior as $Ntoinfty$ of averages of nonconventional arrays having the form $N^{-1}sum_{n=1}^Nprod_{j=1}^ell T^{P_j(n,N)}f_j$ 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 $Ninmathcal N_Lambda$ 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." @default.
- W2949492065 created "2019-06-27" @default.
- W2949492065 creator A5064396762 @default.
- W2949492065 date "2017-02-18" @default.
- W2949492065 modified "2023-09-24" @default.
- W2949492065 title "Ergodic Theorems for Nonconventional Arrays and an Extension of the Szemeredi Theorem" @default.
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- W2949492065 doi "https://doi.org/10.48550/arxiv.1702.05628" @default.
- W2949492065 hasPublicationYear "2017" @default.
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