{
  "id": "1704.07210",
  "version": "v1",
  "published": "2017-04-24T13:30:06.000Z",
  "updated": "2017-04-24T13:30:06.000Z",
  "title": "An improved bound on the Hausdorff dimension of Besicovitch sets in $\\mathbb{R}^3$",
  "authors": [
    "Nets Hawk Katz",
    "Joshua Zahl"
  ],
  "comment": "57 pages, 0 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove that any Besicovitch set in $\\mathbb{R}^3$ must have Hausdorff dimension at least $5/2+\\epsilon_0$ for some small constant $\\epsilon_0>0$. This follows from a more general result about the volume of unions of tubes that satisfy the Wolff axioms. Our proof grapples with a new \"almost counter example\" to the Kakeya conjecture, which we call the $SL_2$ example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension $5/2$. We believe this example may be an interesting object for future study.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-24T13:30:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "besicovitch set",
      "small constant",
      "object resembles",
      "general result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}