{
  "id": "1903.12093",
  "version": "v1",
  "published": "2019-03-28T16:15:28.000Z",
  "updated": "2019-03-28T16:15:28.000Z",
  "title": "On Hausdorff dimension of radial projections",
  "authors": [
    "Bochen Liu"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "For any $x\\in\\mathbb{R}^d$, $d\\geq 2$, denote $\\pi^x: \\mathbb{R}^d\\backslash\\{x\\}\\rightarrow S^{d-1}$ as the radial projection $$\\pi^x(y)=\\frac{y-x}{|y-x|}. $$ Given a Borel set $E\\subset{\\Bbb R}^d$, $\\dim_{\\mathcal{H}} E\\leq d-1$, in this paper we investigate for how many $x\\in \\mathbb{R}^d$ the radial projection $\\pi^x$ preserves the Hausdorff dimension of $E$, namely whether $\\dim_{\\mathcal{H}}\\pi^x(E)=\\dim_{\\mathcal{H}} E$. We develop a general framework to link $\\pi^x(E)$, $x\\in F$ and $\\pi^y(F)$, $y\\in E$, for any Borel set $F\\subset\\mathbb{R}^d$. In particular, whether $\\dim_{\\mathcal{H}}\\pi^x(E)=\\dim_{\\mathcal{H}}E$ for some $x\\in F$ can be reduced to whether $F$ is visible from some $y\\in E$ (i.e. $\\mathcal{H}^{d-1}(\\pi^y(F))>0$). This allows us to apply Orponen's estimate on visibility to obtain $$\\dim_{\\mathcal{H}}\\left\\{x\\in\\mathbb{R}^d: \\dim_{\\mathcal{H}}\\pi^x(E)<\\dim_{\\mathcal{H}}E\\right\\}\\leq 2(d-1)-\\dim_{\\mathcal{H}}E,$$ for any Borel set $E\\subset{\\Bbb R}^d$, $\\dim_{\\mathcal{H}} E\\in(d-2, d-1]$. This improves the Peres-Schlag bound when $\\dim_{\\mathcal{H}} E\\in(d-\\frac{3}{2}, d-1]$, and it is optimal at the endpoint $\\dim_{\\mathcal{H}} E=d-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-28T16:15:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "radial projection",
      "hausdorff dimension",
      "borel set",
      "general framework",
      "apply orponens estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}