| In 1975 Szemeredi proved the long-standing conjecture of Erdo&huml;s and Turan that any subset of Z having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemeredi's proof was entirely combinatorial, but two years later Furstenberg gave a quite different proof of Szemeredi's Theorem by first showing its equivalence to an ergodic-theoretic assertion of multiple recurrence, and then bringing new machinery in ergodic theory to bear on proving that. His ergodic-theoretic approach subsequently yielded several other results in extremal combinatorics, as well as revealing a range of new phenomena according to which the structures of probability-preserving systems can be described and classified.;In this dissertation I report on some recent advances in understanding these ergodic-theoretic structures. It contains new proofs of the norm convergence of the 'nonconventional' ergodic averages that underly Furstenberg's approach to variants of Szemeredi's Theorem, and of two of the recurrence theorems of Furstenberg and Katznelson: the Multidimensional Multiple Recurrence Theorem, which implies a multidimensional generalization of Szemeredi's Theorem; and a density version of the Hales-Jewett Theorem of Ramsey Theory... |
