{
  "id": "2010.11556",
  "version": "v1",
  "published": "2020-10-22T09:29:23.000Z",
  "updated": "2020-10-22T09:29:23.000Z",
  "title": "Dimension of Images of Large Level Sets",
  "authors": [
    "Anthony G. O'Farrell",
    "Gavin Armstrong"
  ],
  "comment": "11 pages, 3 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\\to\\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<\\alpha<1$ let $I_\\alpha(f)$ denote the set of values $y\\in\\mathbb{R}$ whose preimage $f^{-1}(y)$ has Hausdorff dimension $\\dim f^{-1}(y) \\ge \\alpha$. We consider how large can be the Hausdorff dimension of $I_\\alpha(f)$, as $f$ ranges over the set $C^k(E,\\mathbb{R})$ of all $k$-times continuously-differentiable functions from $E$ into $\\mathbb{R}$. We show that the sharp upper bound on $\\dim I_\\alpha(f)$ is $\\displaystyle\\frac{1-\\alpha}k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-22T09:29:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A06",
      "26A18",
      "28A78",
      "37E05",
      "28A80"
    ],
    "keywords": [
      "large level sets",
      "hausdorff dimension",
      "sharp upper bound",
      "natural number",
      "times continuously-differentiable functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}