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

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.