{
  "id": "1801.06384",
  "version": "v1",
  "published": "2018-01-19T12:53:11.000Z",
  "updated": "2018-01-19T12:53:11.000Z",
  "title": "Difference sets and the slice rank bounding method",
  "authors": [
    "Gábor Hegedűs"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $p\\geq 3$ be a prime, $k\\geq 2$ and $n\\geq 1$ be integers. Denote by $Q(k)$ the set of $k$th power residues modulo $p$. Suppose that $1<gcd(k,p-1)<p-1$. Let $A\\subseteq ({\\mathbb Z _p})^n$ be an arbitrary subset such that $$ \\{ \\mathbf{a}-\\mathbf{b}:~\\mathbf{a},\\mathbf{b}\\in A,\\mathbf{a}\\neq \\mathbf{b}\\}\\cap (Q(k))^n=\\emptyset. $$ Then we prove the exponential upper bound $$ |A|\\leq 2\\Big( \\frac{(p-1)(d-1)}{d} + 1 \\Big)^n, $$ where $d=gcd(k,p-1)$. We use in our proof Tao's slice rank bounding method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-19T12:53:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "difference sets",
      "taos slice rank bounding method",
      "th power residues modulo",
      "exponential upper bound",
      "proof taos slice rank bounding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}