{
  "id": "1508.03814",
  "version": "v1",
  "published": "2015-08-16T11:50:05.000Z",
  "updated": "2015-08-16T11:50:05.000Z",
  "title": "Differences of subgroups in subgroups",
  "authors": [
    "Ilya D. Shkredov"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \\ll |\\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \\ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and prove a \"dual\" sum-products estimate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-16T11:50:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "differences",
      "prime field",
      "sufficiently large subgroup",
      "arbitrary multiplicative subgroups",
      "arbitrary set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150803814S"
    }
  }
}