{
  "id": "2205.15101",
  "version": "v1",
  "published": "2022-05-30T13:44:42.000Z",
  "updated": "2022-05-30T13:44:42.000Z",
  "title": "Lower bounds on Bourgain's constant for harmonic measure",
  "authors": [
    "Matthew Badger",
    "Alyssa Genschaw"
  ],
  "comment": "18 pages, 1 figure, 1 table",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "For every $n\\geq 2$, Bourgain's constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\\mathbb{R}^n$ on which harmonic measure is defined. Jones and Wolff (1988) proved that $b_2=1$. When $n\\geq 3$, Bourgain (1987) proved that $b_n>0$ and Wolff (1995) produced examples showing $b_n<1$. Refining Bourgain's original outline, we prove that \\[ b_n\\geq c\\,n^{-2n(n-1)}/\\ln(n)\\] for all $n\\geq 3$, where $c>0$ is a constant that is independent of $n$. We further estimate $b_3\\geq 1\\times 10^{-15}$ and $b_4\\geq 2\\times 10^{-26}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-30T13:44:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B15",
      "28A75",
      "31B25",
      "60J65"
    ],
    "keywords": [
      "harmonic measure",
      "bourgains constant",
      "lower bounds",
      "refining bourgains original outline",
      "largest number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}