{
  "id": "1604.00858",
  "version": "v1",
  "published": "2016-04-04T13:51:31.000Z",
  "updated": "2016-04-04T13:51:31.000Z",
  "title": "Unique expansions and intersections of Cantor sets",
  "authors": [
    "Simon Baker",
    "Derong Kong"
  ],
  "comment": "19 pages, two figures",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "abstract": "To each $\\alpha\\in(1/3,1/2)$ we associate the Cantor set $$\\Gamma_{\\alpha}:=\\Big\\{\\sum_{i=1}^{\\infty}\\epsilon_{i}\\alpha^i: \\epsilon_i\\in\\{0,1\\},\\,i\\geq 1\\Big\\}.$$ In this paper we consider the intersection $\\Gamma_\\alpha \\cap (\\Gamma_\\alpha + t)$ for any translation $t\\in\\mathbb{R}$. We pay special attention to those $t$ with a unique $\\{-1,0,1\\}$ $\\alpha$-expansion, and study the set $$D_\\alpha:=\\{\\dim_H(\\Gamma_\\alpha \\cap (\\Gamma_\\alpha + t)):t \\textrm{ has a unique }\\{-1,0,1\\}\\,\\alpha\\textrm{-expansion}\\}.$$ We prove that there exists a transcendental number $\\alpha_{KL}\\approx 0.39433\\ldots$ such that: $D_\\alpha$ is finite for $\\alpha\\in(\\alpha_{KL},1/2),$ $D_{\\alpha_{KL}}$ is infinitely countable, and $D_{\\alpha}$ contains an interval for $\\alpha\\in(1/3,\\alpha_{KL}).$ We also prove that $D_\\alpha$ equals $[0,\\frac{\\log 2}{-\\log \\alpha}]$ if and only if $\\alpha\\in (1/3,\\frac {3-\\sqrt{5}}{2}].$ As a consequence of our investigation we prove some results on the possible values of $\\dim_{H}(\\Gamma_\\alpha \\cap (\\Gamma_\\alpha + t))$ when $\\Gamma_\\alpha \\cap (\\Gamma_\\alpha + t)$ is a self-similar set. We also give examples of $t$ with a continuum of $\\{-1,0,1\\}$ $\\alpha$-expansions for which we can explicitly calculate $\\dim_{H}(\\Gamma_\\alpha\\cap(\\Gamma_\\alpha+t)),$ and for which $\\Gamma_\\alpha\\cap (\\Gamma_\\alpha+t)$ is a self-similar set. We also construct $\\alpha$ and $t$ for which $\\Gamma_\\alpha \\cap (\\Gamma_\\alpha + t)$ contains only transcendental numbers. Our approach makes use of digit frequency arguments and a lexicographic characterisation of those $t$ with a unique $\\{-1,0,1\\}$ $\\alpha$-expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-04T13:51:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "37B10",
      "37B40",
      "28A78"
    ],
    "keywords": [
      "cantor set",
      "unique expansions",
      "intersection",
      "self-similar set",
      "transcendental number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}