{
  "id": "1001.1980",
  "version": "v2",
  "published": "2010-01-12T19:58:17.000Z",
  "updated": "2010-01-19T16:58:53.000Z",
  "title": "An explicit incidence theorem in F_p",
  "authors": [
    "Harald Andres Helfgott",
    "Misha Rudnev"
  ],
  "comment": "11 pages",
  "doi": "10.1112/S0025579310001208",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $P = A\\times A \\subset \\mathbb{F}_p \\times \\mathbb{F}_p$, $p$ a prime. Assume that $P= A\\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\\mathbb{F}_p$. We show that the pairs of points in $P$ determine $\\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of $n$ points and a set of $n$ lines in the projective plane over $\\F_p$ ($n<\\sqrt{p}$) is bounded by $C n^{{3/2}-{1/10678}}$, where $C$ is an absolute constant.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-01-19T16:58:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B75"
    ],
    "keywords": [
      "explicit incidence theorem",
      "absolute constant",
      "projective plane"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1001.1980A"
    }
  }
}