{
  "id": "2005.02030",
  "version": "v1",
  "published": "2020-05-05T09:54:49.000Z",
  "updated": "2020-05-05T09:54:49.000Z",
  "title": "The weak lower density condition and uniform rectifiability",
  "authors": [
    "Jonas Azzam",
    "Matthew Hyde"
  ],
  "comment": "29 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We show that an Ahlfors $d$-regular set $E$ in $\\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\\in E\\times (0,\\infty)$ for which there exists $y \\in B(x,r)$ and $0<t<r$ satisfying $\\mathscr{H}^{d}_{\\infty}(E\\cap B(y,t))<(2t)^{d}-\\varepsilon(2r)^d$ is a Carleson set for every $\\varepsilon>0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\\varepsilon,\\rho\\in (0,1)$, $A>1$, and $X_{n}$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\\mathscr{B}=\\{B(x,2^{-n}):x\\in X_{n},n\\in \\mathbb{N}\\}$, then \\[ \\sum \\left\\{r_{B}^{s}: B\\in \\mathscr{B}, \\frac{\\mathscr{H}^{s}_{\\rho r_{B}}(X\\cap AB)}{(2r_{B})^{s}}>1+\\varepsilon\\right\\} \\lesssim_{C,A,\\varepsilon,\\rho,s} \\mathscr{H}^{s}(X). \\] This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most $1$ $\\mathscr{H}^{s}$-almost everywhere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-05T09:54:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A78",
      "28A12"
    ],
    "keywords": [
      "weak lower density condition",
      "uniform rectifiability",
      "dimensional hausdorff measure",
      "carleson set",
      "dimensional densities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}