{
  "id": "2203.11609",
  "version": "v1",
  "published": "2022-03-22T10:53:03.000Z",
  "updated": "2022-03-22T10:53:03.000Z",
  "title": "Pointwise convergence in nilmanifolds along smooth functions of polynomial growth",
  "authors": [
    "Konstantinos Tsinas"
  ],
  "comment": "20 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the equidistribution of orbits of the form $b_1^{a_1(n)}... b_k^{a_k(n)}\\Gamma$ in a nilmanifold $X$, where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under certain assumptions on the growth rates of the functions $a_1,...,a_k$, these orbits are uniformly distributed on some subnilmanifold of the space $X$. As an application of these results and in combination with the Host-Kra structure theorem for measure preserving systems, as well as some recent seminorm estimates of the author for ergodic averages concerning Hardy field functions, we deduce a norm convergence result for multiple ergodic averages. Our method mainly relies on an equidistribution result of Green-Tao on finite polynomial orbits of a nilmanifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-22T10:53:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22F30",
      "37A17"
    ],
    "keywords": [
      "polynomial growth",
      "smooth functions",
      "pointwise convergence",
      "nilmanifold",
      "ergodic averages concerning hardy field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}