{
  "id": "2006.06285",
  "version": "v1",
  "published": "2020-06-11T09:44:21.000Z",
  "updated": "2020-06-11T09:44:21.000Z",
  "title": "An improved constant factor for the unit distance problem",
  "authors": [
    "Péter Ágoston",
    "Dömötör Pálvölgyi"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We prove that the number of unit distances among $n$ planar points is at most $1.94\\cdot n^{4/3}$, improving on the previous best bound of $8n^{4/3}$. We also give better upper and lower bounds for several small values of $n$. Our main method is a crossing lemma for multigraphs with a better constant, which is of independent interest, as our proof is simpler than earlier proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-11T09:44:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit distance problem",
      "constant factor",
      "independent interest",
      "planar points",
      "better constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}