{
  "id": "2309.04103",
  "version": "v1",
  "published": "2023-09-08T03:46:50.000Z",
  "updated": "2023-09-08T03:46:50.000Z",
  "title": "New improvement to Falconer distance set problem in higher dimensions",
  "authors": [
    "Xiumin Du",
    "Yumeng Ou",
    "Kevin Ren",
    "Ruixiang Zhang"
  ],
  "comment": "36 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "We show that if a compact set $E\\subset \\mathbb{R}^d$ has Hausdorff dimension larger than $\\frac{d}{2}+\\frac{1}{4}-\\frac{1}{8d+4}$, where $d\\geq 3$, then there is a point $x\\in E$ such that the pinned distance set $\\Delta_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \\ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\\dim_H (E) \\in (\\frac{d}{2} - \\frac{1}{4} - \\frac{3}{8d+4}, \\frac{d}{2}+\\frac{1}{4}-\\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \\ge 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-08T03:46:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78"
    ],
    "keywords": [
      "falconer distance set problem",
      "higher dimensions",
      "pinned distance set",
      "improvement",
      "hausdorff dimension larger"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}