{
  "id": "math/0608595",
  "version": "v1",
  "published": "2006-08-24T02:02:47.000Z",
  "updated": "2006-08-24T02:02:47.000Z",
  "title": "On the distribution of Kloosterman sums",
  "authors": [
    "I. E. Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a prime $p$, we consider Kloosterman sums $$ K_{p}(a) = \\sum_{x\\in \\F_p^*} \\exp(2 \\pi i (x + ax^{-1})/p), \\qquad a \\in \\F_p^*, $$ over a finite field of $p$ elements. It is well known that due to results of Deligne, Katz and Sarnak, the distribution of the sums $K_{p}(a)$ when $a$ runs through $\\F_p^*$ is in accordance with the Sato--Tate conjecture. Here we show that the same holds where $a$ runs through the sums $a = u+v$ for $u \\in \\cU$, $v \\in \\cV$ for any two sufficiently large sets $\\cU, \\cV \\subseteq \\F_p^*$. We also improve a recent bound on the nonlinearity of a Boolean function associated with the sequence of signs of Kloosterman sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-24T02:02:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L05",
      "11L26"
    ],
    "keywords": [
      "kloosterman sums",
      "distribution",
      "boolean function",
      "finite field",
      "sufficiently large sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8595S"
    }
  }
}