{
  "id": "0709.4432",
  "version": "v1",
  "published": "2007-09-27T15:34:21.000Z",
  "updated": "2007-09-27T15:34:21.000Z",
  "title": "On the maximal number of three-term arithmetic progressions in subsets of Z/pZ",
  "authors": [
    "Ben Green",
    "Olof Sisask"
  ],
  "comment": "12 pages",
  "doi": "10.1112/blms/bdn074",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let a be a real number between 0 and 1. Ernie Croot showed that the quantity \\max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided that a is smaller than some absolute constant. In fact we prove rather more, establishing a structure theorem for sets having the maximal number of 3-term progressions amongst all subsets of Z/pZ of cardinality m, provided that m < c*p.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-27T15:34:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "three-term arithmetic progressions",
      "maximal number",
      "real number",
      "ernie croot",
      "absolute constant"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.4432G"
    }
  }
}