{
  "id": "1205.0107",
  "version": "v1",
  "published": "2012-05-01T08:13:30.000Z",
  "updated": "2012-05-01T08:13:30.000Z",
  "title": "Areas of triangles and Beck's theorem in planes over finite fields",
  "authors": [
    "Alex Iosevich",
    "Misha Rudnev",
    "Yujia Zhai"
  ],
  "categories": [
    "math.CO",
    "math.CA",
    "math.NT"
  ],
  "abstract": "It is shown that any subset $E$ of a plane over a finite field $\\F_q$, of cardinality $|E|>q$ determines not less than $\\frac{q-1}{2}$ distinct areas of triangles, moreover once can find such triangles sharing a common base. It is also shown that if $|E|\\geq 64q\\log_2 q$, then there are more than $\\frac{q}{2}$ distinct areas of triangles sharing a common vertex. The result follows from a finite field version of the Beck theorem for large subsets of $\\F_q^2$ that we prove. If $|E|\\geq 64q\\log_2 q$, there exists a point $z\\in E$, such that there are at least $\\frac{q}{4}$ straight lines incident to $z$, each supporting the number of points of $E$ other than $z$ in the interval between $\\frac{|E|}{2q}$ and $\\frac{2|E|}{q}.$ This is proved by combining combinatorial and Fourier analytic techniques. We also discuss higher-dimensional implications of these results in light of recent developments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-01T08:13:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10"
    ],
    "keywords": [
      "becks theorem",
      "distinct areas",
      "fourier analytic techniques",
      "finite field version",
      "straight lines incident"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.0107I"
    }
  }
}