{
  "id": "2406.09707",
  "version": "v1",
  "published": "2024-06-14T04:24:49.000Z",
  "updated": "2024-06-14T04:24:49.000Z",
  "title": "Radial Projections in $\\mathbb{R}^n$ Revisited",
  "authors": [
    "Paige Bright",
    "Yuqiu Fu",
    "Kevin Ren"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given $X,Y\\subset \\mathbb{R}^n$ Borel sets and $X\\neq \\emptyset$. If $\\dim Y \\in (k,k+1]$ for some $k\\in \\{1,\\dots, n-1\\}$, then \\[ \\sup_{x\\in X} \\dim \\pi_x(Y\\setminus \\{x\\}) \\geq \\min \\{\\dim X + \\dim Y - k, k\\}. \\] Our results give a new approach to solving a conjecture of Lund-Pham-Thu in all dimensions and for all ranges of $\\dim Y$. The first of our two methods for proving the above theorem is shorter, utilizing a result of the first author and Gan. Our second method, though longer, follows the original methodology of Orponen--Shmerkin--Wang, and requires a higher dimensional incidence estimate and a dual Furstenberg-set estimate for lines. These new estimates may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-14T04:24:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80"
    ],
    "keywords": [
      "radial projections",
      "higher dimensional incidence estimate",
      "dual furstenberg-set estimate",
      "independent interest",
      "first author"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}