{
  "id": "2205.13890",
  "version": "v1",
  "published": "2022-05-27T10:42:15.000Z",
  "updated": "2022-05-27T10:42:15.000Z",
  "title": "On exceptional sets of radial projections",
  "authors": [
    "Tuomas Orponen",
    "Pablo Shmerkin"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "We prove two new exceptional set estimates for radial projections in the plane. If $K \\subset \\mathbb{R}^{2}$ is a Borel set with $\\dim_{\\mathrm{H}} K > 1$, then $$\\dim_{\\mathrm{H}} \\{x \\in \\mathbb{R}^{2} \\, \\setminus \\, K : \\dim_{\\mathrm{H}} \\pi_{x}(K) \\leq \\sigma\\} \\leq \\max\\{1 + \\sigma - \\dim_{\\mathrm{H}} K,0\\}, \\qquad \\sigma \\in [0,1).$$ If $K \\subset \\mathbb{R}^{2}$ is a Borel set with $\\dim_{\\mathrm{H}} K \\leq 1$, then $$\\dim_{\\mathrm{H}} \\{x \\in \\mathbb{R}^{2} \\, \\setminus \\, K : \\dim_{\\mathrm{H}} \\pi_{x}(K) < \\dim_{\\mathrm{H}} K\\} \\leq 1.$$ The finite field counterparts of both results above were recently proven by Lund, Thang, and Huong Thu. Our results resolve the planar cases of conjectures of Lund-Thang-Huong Thu, and Liu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-27T10:42:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "27A80",
      "28A78"
    ],
    "keywords": [
      "radial projections",
      "borel set",
      "exceptional set estimates",
      "finite field counterparts",
      "lund-thang-huong thu"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}