{
  "id": "2001.08820",
  "version": "v1",
  "published": "2020-01-23T21:33:42.000Z",
  "updated": "2020-01-23T21:33:42.000Z",
  "title": "The metric theory of the pair correlation function of real-valued lacunary sequences",
  "authors": [
    "Niclas Technau",
    "Zeév Rudnick"
  ],
  "comment": "Comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\{ a(x) \\}_{x=1}^{\\infty}$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $\\alpha a(x)$ is Poissonian for Lebesgue almost every $\\alpha\\in \\mathbb{R}$. By using harmonic analysis, our result - irrespective of the choice of the real-valued sequence $\\{ a(x) \\}_{x=1}^{\\infty}$ - can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$ \\vert n_1 (a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2)) \\vert < 1 $$ in integer six-tuples $(n_1,n_2,x_1,x_2,y_1,y_2)$ located in the box $[-N,N]^6$ with the ``excluded diagonals'', that is $$x_1\\neq y_1, \\quad x_2 \\neq y_2, \\quad (n_1,n_2)\\neq (0,0),$$ is at most $N^{4-\\delta}$ for some fixed $\\delta>0$, for all sufficiently large $N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-23T21:33:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J54",
      "11J71"
    ],
    "keywords": [
      "pair correlation function",
      "real-valued lacunary sequences",
      "metric theory",
      "fractional parts",
      "harmonic analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}