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.
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Submitted: June 1997, Revised version: April 1998
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Leibman, A. Multiple Recurrence Theorem for Measure Preserving Actions of a Nilpotent Group. GAFA, Geom. funct. anal. 8, 853–931 (1998). https://doi.org/10.1007/s000390050077
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DOI: https://doi.org/10.1007/s000390050077