{
  "id": "0810.1581",
  "version": "v4",
  "published": "2008-10-09T06:44:46.000Z",
  "updated": "2009-06-29T15:45:20.000Z",
  "title": "Powers of sequences and convergence of ergodic averages",
  "authors": [
    "Nikos Frantzikinakis",
    "Michael Johnson",
    "Emmanuel Lesigne",
    "Mate Wierdl"
  ],
  "comment": "After a few minor corrections, to appear in Ergodic Theory and Dynamical Systems",
  "categories": [
    "math.DS"
  ],
  "abstract": "A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\\mathcal{B},\\mu,T)$ and any bounded measurable function $f$, the averages $ \\frac1N \\sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ is not. Furthermore, we show that for any set of bad exponents $B$, there is a sequence $(s_n)$ where $(s_n^k)$ is good for the mean ergodic theorem exactly when $k$ is not in $B$. We then extend this result to multiple ergodic averages. We also prove a similar result for pointwise convergence of single ergodic averages.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-06-29T15:45:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "28D05",
      "11L15"
    ],
    "keywords": [
      "convergence",
      "multiple ergodic averages",
      "single ergodic averages",
      "invertible measure preserving system",
      "bad exponents"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.1581F"
    }
  }
}