{
  "id": "1608.07344",
  "version": "v1",
  "published": "2016-08-26T00:47:10.000Z",
  "updated": "2016-08-26T00:47:10.000Z",
  "title": "A pathological construction for real functions with large collections of level sets",
  "authors": [
    "Gavin Armstrong"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Consider all the level sets of a real function. We can group these level sets according to their Hausdorff dimensions. We show that the Hausdorff dimension of the collection of all level sets of a given Hausdorff dimension can be arbitrarily close to 1, even if the function is differentiable to some level. By definition of Hausdorff dimension it is clear, for any real function $f(x)$ and any $\\alpha \\in [0,1]$, that $\\dim_{H} \\left\\{ {0.03in} y \\ : \\ \\dim_{H} (f^{-1}(y)) \\geq \\alpha {0.03in} \\right\\} \\leq 1$. What is surprising, and what we show, is that this is actually a sharp bound. That is, $$\\sup \\left\\{ {0.03in} \\dim_{H} \\left\\{ {0.03in} y \\ : \\ \\dim_{H} (f^{-1}(y)) = 1 {0.03in} \\right\\} \\ : \\ f \\in C^{k} {0.03in} \\right\\} = 1,$$ for any $k \\in \\mathbb{Z}_{\\geq 0}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-26T00:47:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A06",
      "26A18",
      "28A78"
    ],
    "keywords": [
      "level sets",
      "real function",
      "hausdorff dimension",
      "large collections",
      "pathological construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}