{
  "id": "0904.0714",
  "version": "v1",
  "published": "2009-04-04T14:21:31.000Z",
  "updated": "2009-04-04T14:21:31.000Z",
  "title": "Pair Correlation for Fractional Parts of $αn^2$",
  "authors": [
    "D. R. Heath-Brown"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We construct real numbers $\\alpha$ for which the pair correlation function \\[N^{-1}#\\{m<n\\le N:||\\alpha m^2-\\alpha n^2||\\le XN^{-1}\\}\\] tends to $X$ as $N$ grows. Moreover we show for any \"Diophantine\" $\\alpha$ that the pair correlation function is $X+O(X^{7/8})+O((\\log N)^{-1}$ for $1\\le X\\le\\log N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-04T14:21:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K31"
    ],
    "keywords": [
      "fractional parts",
      "pair correlation function",
      "construct real numbers",
      "diophantine"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}