### Abstract

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T_{1}, T_{2},..., T_{d}: ℤ {right curved arrow} (X, ∑, μ) ([6]), and so, via the Furstenberg correspondence principle introduced in [5], a new proof of the multi-dimensional Szemerédi Theorem. We bypass the careful manipulation of certain towers of factors of a probability-preserving system that underlies the Furstenberg-Katznelson analysis, instead modifying an approach recently developed in [1] to pass to a large extension of our original system in which this analysis greatly simplifies. The proof is then completed using an adaptation of arguments developed by Tao in [13] for his study of an infinitary analog of the hypergraph removal lemma. In a sense, this addresses the difficulty, highlighted by Tao, of establishing a direct connection between his infinitary, probabilistic approach to the hypergraph removal lemma and the infinitary, ergodic-theoretic approach to Szemerédi's Theorem set in motion by Furstenberg [5].

Original language | English (US) |
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Pages (from-to) | 131-150 |

Number of pages | 20 |

Journal | Journal d'Analyse Mathematique |

Volume | 111 |

Issue number | 1 |

DOIs | |

State | Published - Dec 28 2010 |

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### ASJC Scopus subject areas

- Analysis
- Mathematics(all)

### Cite this

*Journal d'Analyse Mathematique*,

*111*(1), 131-150. https://doi.org/10.1007/s11854-010-0014-3