{
  "id": "2106.03338",
  "version": "v1",
  "published": "2021-06-07T05:06:33.000Z",
  "updated": "2021-06-07T05:06:33.000Z",
  "title": "On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane",
  "authors": [
    "Tuomas Orponen",
    "Pablo Shmerkin"
  ],
  "comment": "48 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "Let $0 \\leq s \\leq 1$ and $0 \\leq t \\leq 2$. An $(s,t)$-Furstenberg set is a set $K \\subset \\mathbb{R}^{2}$ with the following property: there exists a line set $\\mathcal{L}$ of Hausdorff dimension $\\dim_{\\mathrm{H}} \\mathcal{L} \\geq t$ such that $\\dim_{\\mathrm{H}} (K \\cap \\ell) \\geq s$ for all $\\ell \\in \\mathcal{L}$. We prove that for $s\\in (0,1)$, and $t \\in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in $\\mathbb{R}^{2}$ is no smaller than $2s + \\epsilon$, where $\\epsilon > 0$ depends only on $s$ and $t$. For $s>1/2$ and $t = 1$, this is an $\\epsilon$-improvement over a result of Wolff from 1999. The same method also yields an $\\epsilon$-improvement to Kaufman's projection theorem from 1968. We show that if $s \\in (0,1)$, $t \\in (s,2]$ and $K \\subset \\mathbb{R}^{2}$ is an analytic set with $\\dim_{\\mathrm{H}} K = t$, then $$\\dim_{\\mathrm{H}} \\{e \\in S^{1} : \\dim_{\\mathrm{H}} \\pi_{e}(K) \\leq s\\} \\leq s - \\epsilon,$$ where $\\epsilon > 0$ only depends on $s$ and $t$. Here $\\pi_{e}$ is the orthogonal projection to $\\mathrm{span}(e)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-07T05:06:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A75",
      "28A78"
    ],
    "keywords": [
      "hausdorff dimension",
      "furstenberg set",
      "orthogonal projection",
      "kaufmans projection theorem",
      "line set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}