{
  "id": "math/0303212",
  "version": "v1",
  "published": "2003-03-17T23:31:35.000Z",
  "updated": "2003-03-17T23:31:35.000Z",
  "title": "Distance sets corresponding to convex bodies",
  "authors": [
    "Mihail N. Kolountzakis"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "Suppose that $K \\subseteq \\RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \\Norm{x}_K = \\sup\\Set{t\\ge 0: x \\notin tK} $$ on $\\RR^d$. Let also $A\\subseteq\\RR^d$ be a measurable set of positive upper density $\\rho$. We show that if the body $K$ is not a polytope, or if it is a polytope with many faces (depending on $\\rho$), then the distance set $$ D_K(A) = \\Set{\\Norm{x-y}_K: x,y\\in A} $$ contains all points $t\\ge t_0$ for some positive number $t_0$. This was proved by Katznelson and Weiss, by Falconer and Marstrand and by Bourgain in the case where $K$ is the Euclidean ball in any dimension. As corollaries we obtain (a) an extension to any dimension of a theorem of Iosevich and \\L aba regarding distance sets with respect to convex bodies of well-distributed sets in the plane, and also (b) a new proof of a theorem of Iosevich, Katz and Tao about the nonexistence of Fourier spectra for smooth convex bodies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-03-17T23:31:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex body",
      "distance sets corresponding",
      "smooth convex bodies",
      "aba regarding distance sets",
      "usual norm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......3212K"
    }
  }
}