{
  "id": "1012.1130",
  "version": "v3",
  "published": "2010-12-06T11:12:46.000Z",
  "updated": "2011-04-16T18:37:36.000Z",
  "title": "Random Sequences and Pointwise Convergence of Multiple Ergodic Averages",
  "authors": [
    "Nikos Frantzikinakis",
    "Emmanuel Lesigne",
    "Mate Wierdl"
  ],
  "comment": "In Version 2, references have been added. In Version 3, a section on general negative results for recurrence and convergence in the case of non commuting transformations has been added",
  "categories": [
    "math.DS",
    "math.PR"
  ],
  "abstract": "We prove pointwise convergence, as $N\\to \\infty$, for the multiple ergodic averages $\\frac{1}{N}\\sum_{n=1}^N f(T^nx)\\cdot g(S^{a_n}x)$, where $T$ and $S$ are commuting measure preserving transformations, and $a_n$ is a random version of the sequence $[n^c]$ for some appropriate $c>1$. We also prove similar mean convergence results for averages of the form $\\frac{1}{N}\\sum_{n=1}^N f(T^{a_n}x)\\cdot g(S^{a_n}x)$, as well as pointwise results when $T$ and $S$ are powers of the same transformation. The deterministic versions of these results, where one replaces $a_n$ with $[n^c]$, remain open, and we hope that our method will indicate a fruitful way to approach these problems as well.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-04-16T18:37:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiple ergodic averages",
      "pointwise convergence",
      "random sequences",
      "similar mean convergence results",
      "random version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.1130F"
    }
  }
}