Multiple ergodic averages in abelian groups and Khintchine type recurrence

Abstract

Let $G$ be a countable abelian group. We study ergodic averages associated with configurations of the form $\{ag,bg,(a+b)g\}$ for some $a,b\in \mathbb{Z}$. Under some assumptions on $G$, we prove that the universal characteristic factor for these averages is a factor of a $2$-step nilpotent homogeneous space. As an application we derive a Khintchine type recurrence result. In particular, we prove that for every countable abelian group $G$, if $a,b\in \mathbb{Z}$ are such that $aG,bG,(b-a)G$ and $(a+b)G$ are of finite index in $G$, then for every $E\subset G$ and $\epsilon >0$ the set $$\{g\in G:d(E\cap E-ag\cap E-bg\cap E-(a+b)g)\ge d(E{)}^4-\epsilon \}$$ is syndetic. This generalizes previous results for $G=\mathbb{Z}$, $G={\mathbb{F}}_p^{\omega }$ and $G={⨁}_{p\in P}{\mathbb{F}}_p$ by Bergelson Host and Kra, Bergelson Tao and Ziegler and the author, respectively.

37 pages, 1 figure. final accepted version, to appear in Trans. Amer. Math. Soc. Added a structure result for the Conze-Lesigne factor as a double coset, simplified the proofs in section 3 and added various examples

Cited by in corpus (5)

• Multiple recurrence and large intersections for abelian group actions
• Host-Kra factors for ${⨁}_{p\in P}\mathbb{Z}/p\mathbb{Z}$ actions and finite dimensional nilpotent systems
• Khintchine-type double recurrence in abelian groups
• Uniform syndeticity in multiple recurrence
• Multiple recurrence and popular differences for polynomial patterns in rings of integers