{
  "id": "1502.06833",
  "version": "v1",
  "published": "2015-02-24T15:26:36.000Z",
  "updated": "2015-02-24T15:26:36.000Z",
  "title": "Quadratic residues and difference sets",
  "authors": [
    "Vsevolod F. Lev",
    "Jack Sonn"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\\{a+b\\colon a\\in A, b\\in B\\}$ with non-singleton sets $A,B\\subset F_p$. The case $A=B$ of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not of the form $\\{a'-a\"\\colon a',a\"\\in A,\\,a'\\ne a\"\\}$ with $A\\subset F_p$? We attack here a presumably more tractable variant of this problem, which is to show that there is no $A\\subset F_p$ such that every quadratic residue has a \\emph{unique}representation as $a'-a\"$ with $a',a\"\\in A$, and no non-residue is represented in this form. We give a number of necessary conditions for the existence of such $A$, involving for the most part the behavior of primes dividing $p-1$. These conditions enable us to rule out all primes $p$ in the range $13<p<10^{18}$ (the primes $p=5$ and $p=13$ being conjecturally the only exceptions).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-24T15:26:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11A15",
      "11B34",
      "11P70",
      "11T21",
      "05B10"
    ],
    "keywords": [
      "difference sets",
      "quadratic residues modulo",
      "differences remains open",
      "exceptions",
      "sufficiently large primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}