{
  "id": "1402.2799",
  "version": "v4",
  "published": "2014-02-12T12:18:16.000Z",
  "updated": "2014-04-21T10:52:03.000Z",
  "title": "Rectifiability via a square function and Preiss' theorem",
  "authors": [
    "Xavier Tolsa",
    "Tatiana Toro"
  ],
  "comment": "Minor corrections and adjustments",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let $E$ be a set in $\\mathbb R^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that $\\liminf_{r\\to0}r^{-n} H^n(B(x,r)\\cap E)>0$ for $H^n$-a.e. $x\\in E$. In this paper it is shown that $E$ is $n$-rectifiable if and only if $$\\int_0^1 \\left|\\frac{H^n(B(x,r)\\cap E)}{r^n} - \\frac{H^n(B(x,2r)\\cap E)}{(2r)^n}\\right|^2\\,\\frac{dr}r < \\infty$$ for $H^n$-a.e. $x\\in E$; and also if and only if $$ \\lim_{r\\to0}\\left(\\frac{H^n(B(x,r)\\cap E)}{r^n} - \\frac{H^n(B(x,2r)\\cap E)}{(2r)^n}\\right) = 0$$ for $H^n$-a.e. $x\\in E$. Other more general results involving Radon measures are also proved.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-04-21T10:52:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A78",
      "42B20"
    ],
    "keywords": [
      "square function",
      "rectifiability",
      "dimensional hausdorff measure",
      "radon measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.2799T"
    }
  }
}