{
  "id": "math/9903166",
  "version": "v2",
  "published": "1999-03-29T01:43:52.000Z",
  "updated": "2000-09-01T00:00:00.000Z",
  "title": "An improved bound on the Minkowski dimension of Besicovitch sets in R^3",
  "authors": [
    "Nets Hawk Katz",
    "Izabella Łaba",
    "Terence Tao"
  ],
  "comment": "64 pages, published version",
  "journal": "Ann. of Math. (2) 152 (2000), no. 2, 383-446",
  "categories": [
    "math.CA"
  ],
  "abstract": "A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \\R^3. In this paper we show that the Minkowski dimension must in fact be greater than 5/2 + \\epsilon for some absolute constant \\epsilon > 0. One observation arising from the argument is that Besicovitch sets of near-minimal dimension have to satisfy certain strong properties, which we call ``stickiness,'' ``planiness,'' and ``graininess.'' The purpose of this paper is to improve upon the known bounds for the Minkowski dimension of Besicovitch sets in three dimensions. As a by-product of the argument we obtain some strong conclusions on the structure of Besicovitch sets with almost-minimal Minkowski dimension.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2000-09-01T00:00:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B15"
    ],
    "keywords": [
      "besicovitch set",
      "almost-minimal minkowski dimension",
      "unit line segment",
      "hausdorff dimensions",
      "absolute constant"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......3166H"
    }
  }
}