{
  "id": "1006.0491",
  "version": "v2",
  "published": "2010-06-02T20:10:52.000Z",
  "updated": "2010-06-08T19:10:03.000Z",
  "title": "Multiple recurrence and the structure of probability-preserving systems",
  "authors": [
    "Tim Austin"
  ],
  "comment": "81 pages. This was originally submitted as my UCLA Ph.D. dissertation. [TDA Jun 8th 2010:] New version with slight textual corrections in Chapter 6",
  "categories": [
    "math.DS",
    "math.CO",
    "math.PR"
  ],
  "abstract": "In 1975 Szemer\\'edi proved the long-standing conjecture of Erd\\H{o}s and Tur\\'an that any subset of $\\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemer\\'edi's proof was entirely combinatorial, but two years later Furstenberg gave a quite different proof of Szemer\\'edi'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 work I survey some recent advances in understanding these ergodic-theoretic structures. It contains proofs of the norm convergence of the `nonconventional' ergodic averages that underly Furstenberg's approach to variants of Szemer\\'edi's Theorem, and of two of the recurrence theorems of Furstenberg and Katznelson: the Multidimensional Multiple Recurrence Theorem, which implies a multidimensional generalization of Szemer\\'edi's Theorem; and a density version of the Hales-Jewett Theorem of Ramsey Theory.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-06-08T19:10:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28D15",
      "28D05",
      "37A30",
      "37A45",
      "37A50",
      "11L15",
      "60G09",
      "05D10"
    ],
    "keywords": [
      "multiple recurrence",
      "probability-preserving systems",
      "upper banach density contains",
      "arbitrarily long arithmetic progressions",
      "density contains arbitrarily long"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 81,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.0491A"
    }
  }
}