Deducing the multidimensional Szemeredi Theorem from an infinitary 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.