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

Abstract

Let $\mathcal{P}$ be a countable multiset of primes and let $G={⨁}_{p\in P}\mathbb{Z}/p\mathbb{Z}$. 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.

78 pages, 2 figures. Journal version, to appear in Analysis and PDE

References in corpus (2)

• Multiple ergodic averages in abelian groups and Khintchine type recurrence
• Host-Kra theory for ${⨁}_{p\in P}{\mathbb{F}}_p$- Systems and multiple recurrence