{
  "id": "1304.5305",
  "version": "v1",
  "published": "2013-04-19T04:08:02.000Z",
  "updated": "2013-04-19T04:08:02.000Z",
  "title": "On radii of spheres determined by subsets of Euclidean space",
  "authors": [
    "Bochen Liu"
  ],
  "comment": "10 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we consider the problem of how large the Hausdorff dimension of $E\\subset\\R^d$ needs to be in order to ensure that the radii set of $(d-1)$-dimensional spheres determined by $E$ has positive Lebesgue measure. We also study the question of how often can a neighborhood of a given radius repeat. We obtain two results. First, by applying a general mechanism developed in \\cite{mul} for studying Falconer-type problems, we prove that a neighborhood of a given radius cannot repeat more often than the statistical bound if $\\dH(E)>d-1+\\frac{1}{d}$; In $\\R^2$, the dimensional threshold is sharp. Second, by proving an intersection theorem, we prove for a.e $a\\in\\R^d$, the radii set of $(d-1)$-spheres with center $a$ determined by $E$ must have positive Lebesgue measure if $\\dH(E)>d-1$, which is a sharp bound for this problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-19T04:08:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "42B20"
    ],
    "keywords": [
      "euclidean space",
      "positive lebesgue measure",
      "radii set",
      "studying falconer-type problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.5305L"
    }
  }
}