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arXiv · 2409.05087

Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages

Zemer Kosloff·Shrey Sanadhya
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Abstract

We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the $2$-dimensional result to resolve a question of Huang, Shao and Ye and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates.

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Zemer Kosloff, Shrey Sanadhya. 2024-09-08. Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages. https://arxiv.org/abs/2409.05087

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