Multiple Recurrence Theorem for Measure Preserving Actions of a Nilpotent Group

Abstract.

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 $(X, \frak B, \mu$) be a measure space with $\mu (X) \le \infty$ and let $T_1, \dots, T_k$ be measure preserving transformations of X generating a nilpotent group. Then for any $A \in \frak B$ with $\mu (A) \ge 0,$¶¶$\liminf \limits_{N \rightarrow \infty} {1\over N} \sum \limits^{N-1} \limits_{n=0}\mu (T_1^{-n} A \cap \cdots \cap T_k^{-n} A) \ge 0$.¶¶ 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.