Host–Kra factors for ⊕ p∈Pℤ∕pℤ actions and finite-dimensional nilpotent systems

Shalom, Or

doi:10.2140/apde.2024.17.2379

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

Or Shalom

Vol. 17 (2024), No. 7, 2379–2449

DOI: 10.2140/apde.2024.17.2379

Abstract

Let $𝒫$ be a countable multiset of primes and let $G = \oplus_{p \in P} ℤ ∕ p ℤ$ . 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. As an application of our structure theorem 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. Our results provide a counterpart of the structure theorem of Host and Kra (2005) and Ziegler (2007) concerning $ℤ$ -actions and generalize the results of Bergelson, Tao and Ziegler (2011, 2015) concerning $𝔽_{p}^{ω}$ -actions. This is also the first instance of studying the Host–Kra factors of nonfinitely generated groups of unbounded torsion.

Keywords

universal characteristic factors, nilsystems, Gowers–Host–Kra seminorms

Mathematical Subject Classification

Primary: 37A05

Milestones

Received: 27 March 2022

Revised: 14 April 2023

Accepted: 6 June 2023

Published: 21 August 2024

Authors

Or Shalom
Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmond J. Safra Campus Jerusalem Israel

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Vol. 17, No. 7, 2024