{
  "id": "1407.2397",
  "version": "v2",
  "published": "2014-07-09T09:16:52.000Z",
  "updated": "2014-08-17T12:34:52.000Z",
  "title": "Elementary methods for incidence problems in finite fields",
  "authors": [
    "Javier Cilleruelo",
    "Alex Iosevich",
    "Ben Lund",
    "Oliver Roche-Newton",
    "Misha Rudnev"
  ],
  "comment": "9 pages. In this new version, Theorem 3 has been significantly improved, whilst the proof has been simplified. Also, Ben Lund has been added as an author",
  "categories": [
    "math.CO"
  ],
  "abstract": "We use elementary methods to prove an incidence theorem for points and spheres in $\\mathbb{F}_q^n$. As an application, we show that any point set of $P\\subset \\mathbb{F}_q^2$ with $|P|\\geq 5q$ determines a positive proportion of all circles. The latter result is an analogue of Beck's Theorem for circles which is optimal up to multiplicative constants.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-17T12:34:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10"
    ],
    "keywords": [
      "elementary methods",
      "incidence problems",
      "finite fields",
      "incidence theorem",
      "becks theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2397C"
    }
  }
}