Deducing the Multidimensional Szemerédi Theorem from the Infinitary Hypergraph Removal Lemma

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T1, T2, . . . , Td : Z y (X,Σ, μ) ([5]), and so, via the Furstenberg correspondence principle introduced in [4], a new proof of the multi-dimensional Szemerédi Theorem. We bypass the detailed analysis of certain towers of factors of a probability-preserving systems 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. In particular, we find that this simplifies our setting quite quickly to data that can be analyzed using the infinitary version of the hypergraph removal lemma studied by Tao in [11], and we complete the proof by a simple application of that lemma. 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 [4]. CONTENTS