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

Or Shalom

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arXiv · 2006.06470

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

Or Shalom

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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.

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Or Shalom. 2020-06-10. $\bigoplus_{p\in P}\mathbb{F}_p$-Systems as Abramov Systems. https://arxiv.org/abs/2006.06470

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