{
  "id": "2305.11587",
  "version": "v1",
  "published": "2023-05-19T10:55:50.000Z",
  "updated": "2023-05-19T10:55:50.000Z",
  "title": "On the Hausdorff dimension of circular Furstenberg sets",
  "authors": [
    "Katrin Fässler",
    "Jiayin Liu",
    "Tuomas Orponen"
  ],
  "comment": "73 pages, 5 figures",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "For $0 \\leq s \\leq 1$ and $0 \\leq t \\leq 3$, a set $F \\subset \\mathbb{R}^{2}$ is called a circular $(s,t)$-Furstenberg set if there exists a family of circles $\\mathcal{S}$ of Hausdorff dimension $\\dim_{\\mathrm{H}} \\mathcal{S} \\geq t$ such that $$\\dim_{\\mathrm{H}} (F \\cap S) \\geq s, \\qquad S \\in \\mathcal{S}.$$ We prove that if $0 \\leq t \\leq s \\leq 1$, then every circular $(s,t)$-Furstenberg set $F \\subset \\mathbb{R}^{2}$ has Hausdorff dimension $\\dim_{\\mathrm{H}} F \\geq s + t$. The case $s = 1$ follows from earlier work of Wolff on circular Kakeya sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-19T10:55:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78"
    ],
    "keywords": [
      "circular furstenberg sets",
      "hausdorff dimension",
      "circular kakeya sets",
      "earlier work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 73,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}