{
  "id": "1110.3192",
  "version": "v1",
  "published": "2011-10-14T12:53:22.000Z",
  "updated": "2011-10-14T12:53:22.000Z",
  "title": "Intersections of homogeneous Cantor sets and beta-expansions",
  "authors": [
    "Derong Kong",
    "Wenxia Li",
    "Michel Dekking"
  ],
  "comment": "23 pages, 4 figures",
  "journal": "Nonlinearity 23 (2010) 2815-2834",
  "doi": "10.1088/0951-7715/23/11/005",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\Gamma_{\\beta,N}$ be the $N$-part homogeneous Cantor set with $\\beta\\in(1/(2N-1),1/N)$. Any string $(j_\\ell)_{\\ell=1}^\\N$ with $j_\\ell\\in\\{0,\\pm 1,...,\\pm(N-1)\\}$ such that $t=\\sum_{\\ell=1}^\\N j_\\ell\\beta^{\\ell-1}(1-\\beta)/(N-1)$ is called a code of $t$. Let $\\mathcal{U}_{\\beta,\\pm N}$ be the set of $t\\in[-1,1]$ having a unique code, and let $\\mathcal{S}_{\\beta,\\pm N}$ be the set of $t\\in\\mathcal{U}_{\\beta,\\pm N}$ which make the intersection $\\Gamma_{\\beta,N}\\cap(\\Gamma_{\\beta,N}+t)$ a self-similar set. We characterize the set $\\mathcal{U}_{\\beta,\\pm N}$ in a geometrical and algebraical way, and give a sufficient and necessary condition for $t\\in\\mathcal{S}_{\\beta,\\pm N}$. Using techniques from beta-expansions, we show that there is a critical point $\\beta_c\\in(1/(2N-1),1/N)$, which is a transcendental number, such that $\\mathcal{U}_{\\beta,\\pm N}$ has positive Hausdorff dimension if $\\beta\\in(1/(2N-1),\\beta_c)$, and contains countably infinite many elements if $\\beta\\in(\\beta_c,1/N)$. Moreover, there exists a second critical point $\\alpha_c=\\big[N+1-\\sqrt{(N-1)(N+3)}\\,\\big]/2\\in(1/(2N-1),\\beta_c)$ such that $\\mathcal{S}_{\\beta,\\pm N}$ has positive Hausdorff dimension if $\\beta\\in(1/(2N-1),\\alpha_c)$, and contains countably infinite many elements if $\\beta\\in[\\alpha_c,1/N)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-14T12:53:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78"
    ],
    "keywords": [
      "intersection",
      "beta-expansions",
      "contains countably infinite",
      "positive hausdorff dimension",
      "part homogeneous cantor set"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Nonlinearity",
      "year": 2010,
      "month": "Nov",
      "volume": 23,
      "number": 11,
      "pages": 2815
    },
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010Nonli..23.2815K"
    }
  }
}