{
  "id": "1507.00855",
  "version": "v1",
  "published": "2015-07-03T09:24:57.000Z",
  "updated": "2015-07-03T09:24:57.000Z",
  "title": "$\\mathfrak{B}$-free integers in number fields and dynamics",
  "authors": [
    "Aurelia Bartnicka",
    "Joanna Kułaga-Przymus"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Recently, Sarnak initiated the study of the dynamics of the system determined by the square of the M\\\"obius function (the characteristic function of the square-free integers). We deal with his program in the more general context of $\\mathfrak{B}$-free integers in number fields, suggested by Baake and Huck. This setting encompasses the classical square-free case and its generalizations. Given a~number field $K$, let $\\mathfrak{B}$ be a~family of pairwise coprime ideals in its ring of integers $\\mathcal{O}_K$, such that $\\sum_{\\mathfrak{b} \\in \\mathfrak{B}} 1/|\\mathcal{O}_K / \\mathfrak{b}| < \\infty$. We study the dynamical system determined by the set $\\mathcal{F}_\\mathfrak{B} = \\mathcal{O}_K \\setminus \\bigcup_{\\mathfrak{b} \\in \\mathfrak{B}}\\mathfrak{b}$ of $\\mathfrak{B}$-free integers in $\\mathcal{O}_K$. We show that the characteristic function $\\mathbb{1}_{\\mathcal{F}_\\mathfrak{B}}$ of $\\mathcal{F}_\\mathfrak{B}$ is generic for a~probability measure on $\\{0,1\\}^{\\mathcal{O}_K}$, invariant under the corresponding multidimensional shift. The corresponding measure-theoretical dynamical system is proved to be isomorphic to an ergodic rotation on a~compact Abelian group. In particular, it is of zero Kolmogorov entropy. Moreover, we provide a~description of \"patterns\" appearing in $\\mathcal{F}_\\mathfrak{B}$ and compute the topological entropy of the topological system given by the closure of the orbit of $\\mathbb{1}_{\\mathcal{F}_\\mathfrak{B}}$. Finally, we show that this topological dynamical system is proximal and therefore has no non-trivial equicontinuous factor, but has a~non-trivial topological joining with an ergodic rotation on a~compact Abelian group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-03T09:24:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35",
      "37A45",
      "37C85",
      "28D15",
      "37B40",
      "37B05",
      "11N25",
      "11R04",
      "11H06",
      "52C07"
    ],
    "keywords": [
      "free integers",
      "number fields",
      "dynamical system",
      "characteristic function",
      "ergodic rotation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150700855B"
    }
  }
}