{
  "id": "1701.06773",
  "version": "v1",
  "published": "2017-01-24T09:03:31.000Z",
  "updated": "2017-01-24T09:03:31.000Z",
  "title": "Digit frequencies and self-affine sets with non-empty interior",
  "authors": [
    "Simon Baker"
  ],
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\\beta\\in(1,1.787\\ldots)$ then every $x\\in(0,\\frac{1}{\\beta-1})$ has a simply normal $\\beta$-expansion. We also prove that if $\\beta\\in(1,\\frac{1+\\sqrt{5}}{2})$ then every $x\\in(0,\\frac{1}{\\beta-1})$ has a $\\beta$-expansion for which the digit frequency does not exist, and a $\\beta$-expansion with limiting frequency of zeros $p$, where $p$ is any real number sufficiently close to $1/2$. For a class of planar self-affine sets we show that if the horizontal contraction lies in a certain parameter space and the vertical contractions are sufficiently close to $1,$ then every nontrivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and give rise to new examples of self-affine sets with non-empty interior. One particular strength of our approach is that it allows for different rates of contraction in the vertical direction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-24T09:03:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "28A80",
      "11K55"
    ],
    "keywords": [
      "non-empty interior",
      "digit frequency",
      "non-integer bases",
      "nontrivial vertical fibre contains",
      "study digit frequencies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}