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Abstract: In 1975 Szemer\-edi proved the long-standing conjecture of Erd\H{o}s andTur\-an that any subset of $\bbZ$ having positive upper Banach density containsarbitrarily long arithmetic progressions. Szemer\-edi-s proof was entirelycombinatorial, but two years later Furstenberg gave a quite different proof ofSzemer\-edi-s Theorem by first showing its equivalence to an ergodic-theoreticassertion of multiple recurrence, and then bringing new machinery in ergodictheory to bear on proving that. His ergodic-theoretic approach subsequentlyyielded several other results in extremal combinatorics, as well as revealing arange of new phenomena according to which the structures ofprobability-preserving systems can be described and classified.In this work I survey some recent advances in understanding theseergodic-theoretic structures. It contains proofs of the norm convergence of the`nonconventional- ergodic averages that underly Furstenberg-s approach tovariants of Szemer\-edi-s Theorem, and of two of the recurrence theorems ofFurstenberg and Katznelson: the Multidimensional Multiple Recurrence Theorem,which implies a multidimensional generalization of Szemer\-edi-s Theorem; and adensity version of the Hales-Jewett Theorem of Ramsey Theory.



Autor: Tim Austin

Fuente: https://arxiv.org/



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