{
  "id": "1107.4679",
  "version": "v1",
  "published": "2011-07-23T11:04:32.000Z",
  "updated": "2011-07-23T11:04:32.000Z",
  "title": "Average estimate for additive energy in prime field",
  "authors": [
    "Alexey Glibichuk"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Assume that $A\\subseteq \\Fp, B\\subseteq \\Fp^{*}$, $\\1/4\\leqslant\\frac{|B|}{|A|},$ $|A|=p^{\\alpha}, |B|=p^{\\beta}$. We will prove that for $p\\geqslant p_0(\\beta)$ one has $$\\sum_{b\\in B}E_{+}(A, bA)\\leqslant 15 p^{-\\frac{\\min\\{\\beta, 1-\\alpha\\}}{308}}|A|^3|B|.$$ Here $E_{+}(A, bA)$ is an additive energy between subset $A$ and it's multiplicative shift $bA$. This improves previously known estimates of this type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-23T11:04:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T23"
    ],
    "keywords": [
      "additive energy",
      "prime field",
      "average estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.4679G"
    }
  }
}