{
  "id": "1411.0508",
  "version": "v1",
  "published": "2014-11-03T14:44:42.000Z",
  "updated": "2014-11-03T14:44:42.000Z",
  "title": "Convergence of ergodic averages for many group rotations",
  "authors": [
    "Zoltan Buczolich",
    "Gabriella Keszthelyi"
  ],
  "categories": [
    "math.DS",
    "math.CA",
    "math.GR"
  ],
  "abstract": "Suppose that G is a compact Abelian topological group, m is the Haar measure on G and f is a measurable function. Given (n_k), a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages M_N^{\\alpha}f(x). The f-rotation set is Gamma_f={\\alpha \\in G: M_N^{\\alpha} f(x) converges for m a.e. x as N\\to \\infty .} We prove that if G is a compact locally connected Abelian group and f: G -> R is a measurable function then from m(Gamma_f)>0 it follows that f \\in L^1(G). A similar result is established for ordinary Birkhoff averages if G=Z_{p}, the group of p-adic integers. However, if the dual group, \\hat{G} contains \"infinitely many multiple torsion\" then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, f(x+n_{k} {\\alpha})/k, k=1,... for a.e. x for many \\alpha, hence some of our theorems are stated by using instead of Gamma_f slightly larger sets, denoted by Gamma_{f,b}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-03T14:44:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22D40",
      "37A30",
      "28D99",
      "43A40"
    ],
    "keywords": [
      "group rotations",
      "ergodic averages",
      "convergence",
      "non-conventional birkhoff averages",
      "compact locally connected abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}