{
  "id": "1705.06581",
  "version": "v1",
  "published": "2017-05-18T13:20:26.000Z",
  "updated": "2017-05-18T13:20:26.000Z",
  "title": "Products of Differences over Arbitrary Finite Fields",
  "authors": [
    "Brendan Murphy",
    "Giorgis Petridis"
  ],
  "comment": "45 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "There exists an absolute constant $\\delta > 0$ such that for all $q$ and all subsets $A \\subseteq \\mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \\delta}$, then \\[ |(A-A)(A-A)| = |\\{ (a -b) (c-d) : a,b,c,d \\in A\\}| > \\frac{q}{2}. \\] Any $\\delta < 1/13,542$ suffices for sufficiently large $q$. This improves the condition $|A| > q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions. Our proof is based on a qualitatively optimal characterisation of sets $A,X \\subseteq \\mathbb{F}_q$ for which the number of solutions to the equation \\[ (a_1-a_2) = x (a_3-a_4) \\, , \\; a_1,a_2, a_3, a_4 \\in A, x \\in X \\] is nearly maximum. A key ingredient is determining exact algebraic structure of sets $A, X$ for which $|A + XA|$ is nearly minimum, which refines a result of Bourgain and Glibichuk using work of Gill, Helfgott, and Tao. We also prove a stronger statement for \\[ (A-B)(C-D) = \\{ (a -b) (c-d) : a \\in A, b \\in B, c \\in C, d \\in D\\} \\] when $A,B,C,D$ are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-18T13:20:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30"
    ],
    "keywords": [
      "arbitrary finite fields",
      "differences",
      "determining exact algebraic structure",
      "prime field",
      "qualitatively optimal characterisation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}