{
  "id": "1610.00511",
  "version": "v1",
  "published": "2016-10-03T12:19:36.000Z",
  "updated": "2016-10-03T12:19:36.000Z",
  "title": "Ergodic averages with prime divisor weights in $L^{1}$",
  "authors": [
    "Zoltan Buczolich"
  ],
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "We show that $ { \\omega }(n)$ and $ { \\Omega }(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\\sum_{n\\leq K}g(n)$ then for every ergodic dynamical system $(X, { { \\cal A } },\\mu, { \\tau })$ and every $f\\in L^{1}(X)$ $$\\lim_{K\\to { \\infty }} \\frac{1}{S_{g,K}}\\sum_{n=1}^{K} g(n)f( { \\tau }^{n}x)=\\int_{X}fd\\mu \\text{ for $\\mu$ a.e. }x\\in X. $$ This answers a question raised by C. Cuny and M. Weber who showed this result for $L^{p}$, $p>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-03T12:19:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "11A25",
      "28D05",
      "37A05"
    ],
    "keywords": [
      "prime divisor weights",
      "ergodic averages",
      "distinct prime factors",
      "ergodic dynamical system",
      "pointwise ergodic theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}