{
  "id": "2002.11607",
  "version": "v1",
  "published": "2020-02-26T16:50:07.000Z",
  "updated": "2020-02-26T16:50:07.000Z",
  "title": "Equidistribution results for self-similar measures",
  "authors": [
    "Simon Baker"
  ],
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\\infty}$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for self-similar measures. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\\infty}$ where $(f_n)_{n=1}^{\\infty}$ is a sequence of sufficiently smooth real valued functions satisfying a nonlinearity assumption. As a corollary of our main result, we show that if $C$ is equal to the middle third Cantor set and $t\\geq 1$, then with respect to the Cantor-Lebesgue measure on $C+t$ the sequence $(x^n)_{n=1}^{\\infty}$ is uniformly distributed for almost every $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-26T16:50:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar measures",
      "equidistribution results",
      "real valued functions satisfying",
      "middle third cantor set",
      "sufficiently smooth real valued functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}