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

TL;DR

This paper extends Host-Kra structure theory to systems with a direct sum of finite fields over a multiset of primes, leading to new multiple recurrence results in ergodic theory.

Contribution

It generalizes the Host-Kra theory for universal characteristic factors to a broader class of abelian groups formed by direct sums over primes.

Findings

Established a generalized Host-Kra structure theory for these systems.

Derived a Khintchine-type recurrence theorem for the new class of systems.

Extended previous results from fixed prime cases to multiset prime configurations.

Abstract

Let $\mathcal{P}$ be an (unbounded) countable multiset of primes (i.e. every prime may appear multiple times) and let $G=\bigoplus_{p\in \mathcal{P}}\mathbb{F}_p$. 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.