{
  "id": "1810.08127",
  "version": "v1",
  "published": "2018-10-18T16:01:45.000Z",
  "updated": "2018-10-18T16:01:45.000Z",
  "title": "Hausdorff dimension of pinned distance sets and the $L^2$-method",
  "authors": [
    "Bochen Liu"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "We prove that for any $E\\subset{\\Bbb R}^2$, $\\dim_{\\mathcal{H}}(E)>1$, there exists $x\\in E$ such that the Hausdorff dimension of the pinned distance set $$\\Delta_x(E)=\\{|x-y|: y \\in E\\}$$ is no less than $\\min\\left\\{\\frac{4}{3}\\dim_{\\mathcal{H}}(E)-\\frac{2}{3}, 1\\right\\}$. This answers a question recently raised by Guth, Iosevich, Ou and Wang, as well as improves results of Keleti and Shmerkin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-18T16:01:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pinned distance set",
      "hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}