{
  "id": "1710.01911",
  "version": "v1",
  "published": "2017-10-05T08:26:12.000Z",
  "updated": "2017-10-05T08:26:12.000Z",
  "title": "A metric theory of minimal gaps",
  "authors": [
    "Zeév Rudnick"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the minimal gap statistic for fractional parts of sequences of the form $\\mathcal A^\\alpha = \\{\\alpha a(n)\\}$ where $\\mathcal A = \\{a(n)\\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\\alpha$, the minimal gap $\\delta_{\\min}^\\alpha(N)=\\min\\{\\alpha a(m)-\\alpha a(n)\\bmod 1: 1\\leq m\\neq n\\leq N\\}$ is close to that of a random sequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-05T08:26:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "metric theory",
      "minimal gap statistic",
      "random sequence",
      "fractional parts",
      "additive energy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}