Multiple recurrence theorem for measure preserving actions of a nilpotent group
\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 \[ S=\left\{ n\in{\mathbb Z}^d\mid \mu\left(\bigcap_{i=1}^{I}A \left(T_t^{p_{i,t}(n)}\dots T_1^{p_{i,1}(n)}\right)^{-1}\right)>c\right\} \] is syndetic in \({\mathbb Z}^d\) (that is, has bounded gaps). The main result is proved using ideas from the work of Furstenberg and Katznelson, in which the system is represented as a transfinite tower of invariant factors, allowing the argument to be reduced to relatively simple ``primitive'' extensions, and ideas from \textit{V. Bergelson} [Ergodic. Theory Dyn. Syst. 7, 337-349 (1987; Zbl 0645.28012)] for handling polynomials. A great deal of non-trivial machinery needs to be developed for the problem at hand.
scientific article; zbMATH DE number 1223301