{
  "id": "2101.02811",
  "version": "v1",
  "published": "2021-01-08T00:47:22.000Z",
  "updated": "2021-01-08T00:47:22.000Z",
  "title": "Multiple recurrence and large intersections for abelian group actions",
  "authors": [
    "Ethan Ackelsberg",
    "Vitaly Bergelson",
    "Andrew Best"
  ],
  "comment": "57 pages",
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $\\varphi, \\psi : G \\to G$ are homomorphisms such that $\\varphi(G)$, $\\psi(G)$, and $(\\psi - \\varphi)(G)$ have finite index in $G$, then for every ergodic measure-preserving system $(X, \\mathcal{B}, \\mu, (T_g)_{g \\in G})$, every set $A \\in \\mathcal{B}$, and every $\\varepsilon > 0$, the set $\\{g \\in G : \\mu(A \\cap T_{\\varphi(g)}^{-1}A \\cap T_{\\psi(g)}^{-1}A) > \\mu(A)^3 - \\varepsilon\\}$ is syndetic. (2) If $G$ is a countable abelian group and $r, s \\in \\mathbb{Z}$ are integers such that $rG$, $sG$, and $(r \\pm s)G$ have finite index in $G$, and no quotient of $G$ is isomorphic to a Pr\\\"{u}fer $p$-group with $p \\mid rs(s-r)$, then for every ergodic measure-preserving system $(X, \\mathcal{B}, \\mu, (T_g)_{g \\in G})$, every set $A \\in \\mathcal{B}$, and every $\\varepsilon > 0$, the set $\\{g \\in G : \\mu(A \\cap T_{rg}^{-1}A \\cap T_{sg}^{-1}A \\cap T_{(r+s)g}^{-1}A) > \\mu(A)^4 - \\varepsilon\\}$ is syndetic. In particular, this extends and generalizes results of Bergelson, Host, and Kra concerning $\\mathbb{Z}$-actions and of Bergelson, Tao, and Ziegler concerning $\\mathbb{F}_p^{\\infty}$-actions. Using an ergodic version of the Furstenberg correspondence principle, we obtain new combinatorial applications. Our results lead to a number of interesting questions and conjectures, formulated in the introduction and at the end of the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-08T00:47:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A15",
      "37A30"
    ],
    "keywords": [
      "abelian group actions",
      "large intersections",
      "multiple recurrence",
      "countable abelian group",
      "ergodic measure-preserving system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}