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We offer a new proof of the Furstenberg-Katznelson multiple recurrencetheorem for several commuting probability-preserving transformations T_1, T_2,>..., T_d: \bbZ\curvearrowright (X,\S,\mu), and so, via the Furstenbergcorrespondence principle introduced in, a new proof of the multi-dimensionalSzemeredi Theorem. We bypass the careful manipulation of certain towers offactors of a probability-preserving system that underlies theFurstenberg-Katznelson analysis, instead modifying an approach recentlydeveloped for the study of convergence of nonconventional ergodic averages topass to a large extension of our original system in which this analysis greatlysimplifies. The proof is then completed using an adaptation of argumentsdeveloped by Tao for his study of an infinitary analog of the hypergraphremoval lemma. In a sense, this addresses the difficulty, highlighted by Tao,of establishing a direct connection between his infinitary, probabilisticapproach to the hypergraph removal lemma and the infinitary, ergodic-theoreticapproach to Szemeredi's Theorem set in motion by Furstenberg.

Deducing the multidimensional Szemerédi theorem from an infinitary removal lemma