{
  "id": "2008.01678",
  "version": "v1",
  "published": "2020-08-04T16:27:23.000Z",
  "updated": "2020-08-04T16:27:23.000Z",
  "title": "Distinct distances on hyperbolic surfaces",
  "authors": [
    "Xianchang Meng"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For any cofinite Fuchsian group $\\Gamma\\subset {\\rm PSL}(2, \\mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $\\Gamma\\backslash\\mathbb{H}^2$ determines $\\geq C_{\\Gamma} \\frac{N}{\\log N}$ distinct distances for some constant $C_{\\Gamma}>0$ depending only on $\\Gamma$. In particular, for $\\Gamma$ being any finite index subgroup of ${\\rm PSL}(2, \\mathbb{Z})$ with $\\mu=[{\\rm PSL}(2, \\mathbb{Z}): \\Gamma ]<\\infty$, any set of $N$ points on $\\Gamma\\backslash\\mathbb{H}^2$ determines $\\geq C\\frac{N}{\\mu\\log N}$ distinct distances for some absolute constant $C>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T16:27:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10",
      "11P21",
      "11F06"
    ],
    "keywords": [
      "distinct distances",
      "hyperbolic surface",
      "finite index subgroup",
      "cofinite fuchsian group",
      "determines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}