{
  "id": "2201.09641",
  "version": "v1",
  "published": "2022-01-24T12:52:08.000Z",
  "updated": "2022-01-24T12:52:08.000Z",
  "title": "On the dimension of certain sets araising in the base two expansion",
  "authors": [
    "Jörg Neunhäuserer"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that for the base two expansion \\[ x=\\sum_{i=1}^{\\infty}2^{-(d_{1}(x)+d_{2}(x)+\\dots+d_{i}(x))}\\] with $x\\in(0,1]$ and $d_{i}(x)\\in\\mathbb{N}$ the set $A=\\{x|\\lim_{i\\to\\infty}d_{i}(x)=\\infty\\}$ has Hausdorff dimension zero and set $B=\\{x|\\limsup_{i\\to\\infty}d_{i}(x)=\\infty\\}$ has Hausdorff dimension one. This result is strongly opposed to a result on the continued fraction expansion, here both sets have Hausdorff dimension $1/2$, see \\cite{[GO],[LU]}. In addition we will find a dimension spectrum with respect to the base two expansion in the set $B$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-24T12:52:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A80"
    ],
    "keywords": [
      "sets araising",
      "hausdorff dimension zero",
      "continued fraction expansion",
      "dimension spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}