{
  "id": "1810.07083",
  "version": "v1",
  "published": "2018-10-16T15:36:25.000Z",
  "updated": "2018-10-16T15:36:25.000Z",
  "title": "On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne",
  "authors": [
    "Simon Baker",
    "Derong Kong"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $F=\\{\\mathbf{p}_0,\\ldots,\\mathbf{p}_n\\}$ be a collection of points in $\\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\\mathbf{x})=\\lambda \\mathbf{x} +(1-\\lambda)\\mathbf{p}_i,$$ where $\\lambda \\in(0,1)$. Given $F$ and $\\lambda$ it is well known that there exists a unique non-empty compact set $X$ satisfying $X=\\cup_{i=0}^n S_i(X)$. For each $\\mathbf{x} \\in X$ there exists a sequence $\\mathbf{a}\\in\\{0,\\ldots,n\\}^{\\mathbb{N}}$ satisfying $$\\mathbf{x}=\\lim_{j\\to\\infty}(S_{a_1}\\circ \\cdots \\circ S_{a_j})(\\mathbf{0}).$$ We call such a sequence a coding of $\\mathbf{x}$. In this paper we prove that for any $F$ and $k \\in\\mathbb{N},$ there exists $\\delta_k(F)>0$ such that if $\\lambda\\in(1-\\delta_k(F),1),$ then every point in the interior of $X$ has a coding which is $k$-simply normal. Similarly, we prove that there exists $\\delta_{uni}(F)>0$ such that if $\\lambda\\in(1-\\delta_{uni}(F),1),$ then every point in the interior of $X$ has a coding containing all finite words. For some specific choices of $F$ we obtain lower bounds for $\\delta_k(F)$ and $\\delta_{uni}(F)$. We also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction. Our proofs rely on a variation of a well known construction of a normal number due to Champernowne, and an approach introduced by Erd\\H{o}s and Komornik.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-16T15:36:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "11K16",
      "11K55"
    ],
    "keywords": [
      "self-similar sets",
      "construction",
      "champernowne",
      "iterated function systems",
      "complexity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}