$\bigoplus_{p\in P}\mathbb{F}_p$-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.