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Let $\mathcal{P}$ be an (unbounded) countable multiset of primes (i.e. everyprime may appear multiple times) and let $G=\bigoplus_{p\in\mathcal{P}}\mathbb{F}_p$. We develop a Host-Kra structure theory for theuniversal characteristic factors of an ergodic $G$-system. More specifically,we generalize the main results of Bergelson Tao and Ziegler who studied thesefactors in the special case $\mathcal{P}=\{p,p,p,...\}$ for some fixed prime$p$. As an application we deduce a Khintchine-type recurrence theorem in theflavor of Bergelson Tao and Ziegler and Bergelson Host and Kra.

Host-Kra theory for $\bigoplus_{p\in P}\mathbb{F}_p$- Systems and multiple recurrence