{
  "id": "1110.6805",
  "version": "v1",
  "published": "2011-10-31T14:31:14.000Z",
  "updated": "2011-10-31T14:31:14.000Z",
  "title": "On the Mattila-Sjolin theorem for distance sets",
  "authors": [
    "Alex Iosevich",
    "Mihalis Mourgoglou",
    "Krystal Taylor"
  ],
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \\subset {\\Bbb R}^d$, $d \\ge 2$, is greater than $\\frac{d+1}{2}$, then the distance set $\\Delta(E)=\\{|x-y|: x,y \\in E \\}$ contains an interval. We prove this result for distance sets $\\Delta_B(E)=\\{{||x-y||}_B: x,y \\in E \\}$, where ${|| \\cdot ||}_B$ is the metric induced by the norm defined by a symmetric bounded convex body $B$ with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-31T14:31:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B35",
      "52C10",
      "28A75"
    ],
    "keywords": [
      "distance set",
      "mattila-sjolin theorem",
      "symmetric bounded convex body",
      "compact set",
      "hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6805I"
    }
  }
}