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

Let $\mathcal{P}$ be an (unbounded) countable multiset of primes, let $G={⨁}_{p\in P}{\mathbb{F}}_p$. 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 $P=\{p,p,p,\mathrm{.}\mathrm{.}\mathrm{.}\}$ for some fixed prime $p$. In a subsequent paper we use this result to prove a general structure theorem for ergodic ${⨁}_{p\in P}{\mathbb{F}}_p$-systems.