$\bigoplus_{p\in P}\mathbb{F}_p$-Systems as Abramov Systems

TL;DR

This paper investigates the structure of ergodic systems with actions by direct sums of finite fields, generalizing previous results and identifying conditions under which these systems are generated by phase polynomials.

Contribution

It extends the theory of Abramov systems to more general $igoplus_{p ext{ in } P} ext{F}_p$-systems and provides conditions for their extensions to be generated by phase polynomials.

Findings

Identifies conditions for extensions to be generated by phase polynomials.

Provides an example of a non-Abramov ergodic system.

Generalizes previous results by Bergelson, Tao, and Ziegler.

Abstract

Let $\mathcal{P}$ be an (unbounded) countable multiset of primes, let $G=\bigoplus_{p\in P}\mathbb{F}_p$. We study the $k$'th universal characteristic factors of an ergodic probability system $(X,\mathcal{B},\mu)$ with respect to some measure preserving action of $G$. We find conditions under which every extension of these factors is generated by phase polynomials and we give an example of an ergodic $G$-system that is not Abramov. In particular we generalize the main results of Bergelson Tao and Ziegler who proved a similar theorem in the special case $P=\{p,p,p,...\}$ for some fixed prime $p$. In a subsequent paper we use this result to prove a general structure theorem for ergodic $\bigoplus_{p\in P}\mathbb{F}_p$-systems.