Host-Kra theory for ${⨁}_{p\in P}{\mathbb{F}}_p$- Systems and multiple recurrence

Abstract

Let $\mathcal{P}$ be an (unbounded) countable multiset of primes (i.e. every prime may appear multiple times) and let $G={⨁}_{p\in \mathcal{P}}{\mathbb{F}}_p$. We develop a Host-Kra structure theory for the universal characteristic factors of an ergodic $G$-system. More specifically, we generalize the main results of Bergelson Tao and Ziegler who studied these factors in the special case $\mathcal{P}=\{p,p,p,...\}$ for some fixed prime $p$. As an application we deduce a Khintchine-type recurrence theorem in the flavor of Bergelson Tao and Ziegler and Bergelson Host and Kra.

65 pages. Added an acknowledgement. This is the final accepted version appeared in Ergodic Theory and Dynamical Systems. arXiv admin note: substantial text overlap with arXiv:2006.06470

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