Deducing the multidimensional Szemeredi Theorem from an infinitary removal lemma

TL;DR

This paper presents a new proof of the multidimensional Szemeredi Theorem by combining ergodic theory and an infinitary hypergraph removal lemma, simplifying previous complex factor manipulations.

Contribution

It introduces a novel approach that bypasses intricate factor tower manipulations, linking ergodic theory with an infinitary hypergraph removal lemma for the first time.

Findings

New proof of the multidimensional Szemeredi Theorem

Simplified analysis using large extensions of systems

Established a connection between ergodic theory and hypergraph removal lemma

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: \bbZ\curvearrowright (X,\S,\mu), and so, via the Furstenberg correspondence principle introduced in, a new proof of the multi-dimensional Szemeredi 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 for the study of convergence of nonconventional ergodic averages 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 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 Szemeredi's Theorem set in motion by Furstenberg.