Host-Kra factors for \(\bigoplus_{p \in P} \mathbb{Z} / p \mathbb{Z}\) actions and finite-dimensional nilpotent systems (Q6612316)

The theory of universal characteristics, mean multiple ergodic theorems, and Gowers seminorms is developed here in the setting of actions of a nonfinitely generated group of unbounded torsion. Specifically, for actions of \(G=\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}\), where \(P\) is a countable multiset of primes, it is shown that the universal characteristic factor of order less than \(k+1\) is a factor of an inverse limit of \(k\)-step nilpotent homogeneous spaces analogous to the \(k\)-step nilsystem in the case of a \(\mathbb{Z}\)-action in the theory of \textit{B. Host} and \textit{B. Kra} [Ann. Math. (2) 161, No. 1, 397--488 (2005; Zbl 1077.37002)]. These structural results are used to give a new proof of mean convergence for multiple ergodic averages related to \(k\)-term arithmetic progressions in \(G\). The results generalize the case of \(\mathbb{F}_p^{\omega}\)-actions studied by \textit{V. Bergelson} et al. [J. Anal. Math. 127, 329--378 (2015; Zbl 1361.37006)].