Multiple Recurrence Theorem for Measure Preserving Actions of a Nilpotent Group

The multidimensional ergodic Szemerédi theorem of Furstenberg and Katznelson, which deals with commuting transformations, is extended to the case where the transformations generate a nilpotent group: ¶Theorem. Let\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $(X, \frak B, \mu$\end{document}) be a measure space with\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \mu (X) \le \infty $\end{document}and let\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T_1, \dots, T_k$\end{document}be measure preserving transformations of X generating a nilpotent group. Then for any\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $A \in \frak B$\end{document}with\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mu (A) \ge 0,$\end{document}¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \liminf \limits_{N \rightarrow \infty} {1\over N} \sum \limits^{N-1} \limits_{n=0}\mu (T_1^{-n} A \cap \cdots \cap T_k^{-n} A) \ge 0 $\end{document}.¶¶ Our main result also generalizes the polynomial Szemerédi Theorem in [BL1]. In the course of the proof we describe a relatively simple form to which any unitary action of a finitely generated nilpotent group on a Hilbert space and any measure preserving action of a finitely generated nilpotent group on a probability space can be reduced.