{
  "id": "1306.6543",
  "version": "v1",
  "published": "2013-06-27T15:17:11.000Z",
  "updated": "2013-06-27T15:17:11.000Z",
  "title": "The two-point correlation function of the fractional parts of \\sqrt{n} is Poisson",
  "authors": [
    "Daniel El-Baz",
    "Jens Marklof",
    "Ilya Vinogradov"
  ],
  "comment": "15 pages, 3 figures",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Elkies and McMullen [Duke Math.J.~123 (2004) 95--139] have shown that the gaps between the fractional parts of \\sqrt n for n=1,\\ldots,N, have a limit distribution as N tends to infinity. The limit distribution is non-standard and differs distinctly from the exponential distribution expected for independent, uniformly distributed random variables on the unit interval. We complement this result by proving that the two-point correlation function of the above sequence converges to a limit, which in fact coincides with the answer for independent random variables. We also establish the convergence of moments for the probability of finding r points in a randomly shifted interval of size 1/N. The key ingredient in the proofs is a non-divergence estimate for translates of certain non-linear horocycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-27T15:17:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J71",
      "11K36",
      "11P21",
      "22E40",
      "37A17",
      "37A25"
    ],
    "keywords": [
      "two-point correlation function",
      "fractional parts",
      "limit distribution",
      "independent random variables",
      "uniformly distributed random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6543E"
    }
  }
}