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

TL;DR

This paper extends the structure theorem for universal characteristic factors to actions of groups formed by direct sums of cyclic groups of prime order, revealing their connection to finite dimensional nilpotent systems.

Contribution

It introduces a structure theorem for Gowers-Host-Kra factors in non-finitely generated groups with unbounded torsion, generalizing prior results for $bz$ and $bf_p^$ actions.

Findings

Universal characteristic factors are inverse limits of finite dimensional $k$-step nilpotent systems.

Provides an alternative proof for $L^2$-convergence of multiple ergodic averages.

Derives a formula for limits in nilpotent homogeneous systems.

Abstract

Let $\mathcal{P}$ be a countable multiset of primes and let $G=\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}$. We study the universal characteristic factors associated with the Gowers-Host-Kra seminorms for the group $G$. We show that the universal characteristic factor of order $<k+1$ is a factor of an inverse limit of finite dimensional $k$-step nilpotent homogeneous spaces. The latter is a counterpart of a $k$-step nilsystem where the homogeneous group is not necessarily a Lie group. This result provides a counterpart of the structure theorem of Host-Kra and Ziegler concerning $\mathbb{Z}$-actions and generalizes the results of Bergelson Tao and Ziegler concerning $\mathbb{F}_p^\omega$-actions. This result is the first instance of a structure theorem for the universal characteristic factors associated with a non-finitely generated group of unbounded torsion. As an application we derive an alternative proof for the $L^2$-convergence of multiple ergodic averages associated with $k$-term arithmetic progressions in $G$ and derive a formula for the limit in the special case where the underlying space is a nilpotent homogeneous system.