{
  "id": "1608.05033",
  "version": "v1",
  "published": "2016-08-17T18:08:52.000Z",
  "updated": "2016-08-17T18:08:52.000Z",
  "title": "Fluctuations of Ergodic Averages for Actions of Groups of Polynomial Growth",
  "authors": [
    "Nikita Moriakov"
  ],
  "comment": "15 pages; preliminary, comments are welcome",
  "categories": [
    "math.DS"
  ],
  "abstract": "It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\\mathbb{Z}^d$ on a probability space $X$ and a nonnegative measurable function $f$ on $X$, the probability that the sequence of ergodic averages $$ \\frac 1 {(2k+1)^d} \\sum\\limits_{g \\in [-k,\\dots,k]^d} f(g \\cdot x) $$ has at least $n$ fluctuations across an interval $(\\alpha,\\beta)$ can be bounded from above by $c_1 c_2^n$ for some universal constants $c_1 \\in \\mathbb{R}$ and $c_2 \\in (0,1)$, which depend only on $d,\\alpha,\\beta$. The purpose of this article is to generalize this result to measure-preserving actions of groups of polynomial growth. As the main tool we develop a generalization of effective Vitali covering theorem for groups of polynomial growth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-17T18:08:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28D05",
      "28D15"
    ],
    "keywords": [
      "polynomial growth",
      "ergodic averages",
      "fluctuations",
      "measure-preserving action",
      "probability space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}