Multiple recurrence theorem for measure preserving actions of a nilpotent group
type: Item
instance of: scholarly article
title: Multiple recurrence theorem for measure preserving actions of a nilpotent group
zbMATH Open document ID: 0917.28017
published in: Geometric and Functional Analysis. GAFA
publication date: 1999-07-20T00:00:00Z
review text: \textit{H. Furstenberg} and \textit{Y. Katznelson} [J. Anal. Math. 34, 275-291 (1978; Zbl 0426.28014)] proved a multidimensional version of the ergodic Szemerédi theorem for commuting transformations: if \(T_1,\dots,T_k\) are commuting measure-preserving transformations of the probability space \((X,\mathcal{B},\mu)\), then \[ \liminf_{N\to\infty}{1\over N} \sum_{n=0}^{N-1}\mu\left( T_1^{-n}A\cap\dots\cap T_k^{-n}A\right)>0 \] for any \(A\in\mathcal B\) with \(\mu(A)>0\). This result has many combinatorial consequences, and also implies the earlier topological multiple recurrence theorem for commuting homeomorphisms of \textit{H. Furstenberg} and \textit{B. Weiss} [J. Anal. Math. 34, 61-85 (1978; Zbl 0425.54023], though the topological theorem is an important part of the proof of the measure-theoretic theorem. It is natural to ask if the requirement that the transformations commute can be dropped. Furstenberg has exhibited examples in which \(T_1,\dots,T_k\) generate a metabelian group for which the topological (and hence the measure-theoretic) multiple recurrence theorem fails. On the other hand, \textit{A. Leibman} [Geom. Funct. Anal. 4, No. 6, 648-659 (1994; Zbl 0848.54025)] has shown that topological multiple recurrence holds along polynomials for nilpotent group actions. In this paper the measure-theoretic multiple recurrence theorem is proved for nilpotent group actions along polynomials, as a consequence of a ``uniform multiparameter'' version of the result: let \(G\) be a nilpotent group of measure-preserving transformations of the probability space \((X,\mathcal B,\mu)\) acting on the right, \(T_1,\dots,T_t\) members of \(G\), \(d\) a natural number, \(p_{i,j}:{\mathbb Z}^d\to{\mathbb Z}\) polynomials with \(p_{i,j}(0)=0\) for \(1\leq i\leq I\), \(1\leq j\leq t\). Then for any \(A\) with \(\mu(A)>0\) there is a \(c>0\) for which the set \[ S=\left\{ n\in{\mathbb Z}^d\mid \mu\left(\bigcap_{i=1}^{I}A \left(T_t^{p_{i,t}(n)}\dots T_1^{p_{i,1}(n)}\right)^{-1}\ri…
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 1223301
zbMATH Keywords: multiple recurrence
zbMATH Keywords: polynomial Szemerédi theorem
zbMATH Keywords: nilpotent group actions
zbMATH Keywords: ergodic Szemerédi theorem
zbMATH Keywords: measure-preserving transformations
author: A. Leibman
reviewed by: Thomas B. Ward
MaRDI profile type: Publication
full work available at URL: s000390050077
OpenAlex ID: W2056043123
DOI: 10.1007/S000390050077
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description: scientific article; zbMATH DE number 1223301