{
  "id": "1805.03713",
  "version": "v1",
  "published": "2018-05-09T20:00:19.000Z",
  "updated": "2018-05-09T20:00:19.000Z",
  "title": "Normal numbers and nested perfect necklaces",
  "authors": [
    "Verónica Becher",
    "Olivier Carton"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "M. B. Levin used Sobol-Faure low discrepancy sequences with Pascal matrices modulo $2$ to construct, for each integer $b$, a real number $x$ such that the first $N$ terms of the sequence $(b^n x \\mod 1)_{n\\geq 1}$ have discrepancy $O((\\log N)^2/N)$. This is the lowest discrepancy known for this kind of sequences. In this note we characterize Levin's construction in terms of nested perfect necklaces, which are a variant of the classical de Bruijn necklaces. Moreover, we show that every real number $x$ whose base $b$ expansion is the concatenation of nested perfect necklaces of exponentially increasing order satisfies that the first $N$ terms of $(b^n x \\mod 1)_{n\\geq 1}$ have discrepancy $O((\\log N)^2/N)$. For base $2$ and the order being a power of $2$, we give the exact number of nested perfect necklaces and an explicit method based on matrices to construct each of them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-09T20:00:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R15",
      "11K16",
      "11K38"
    ],
    "keywords": [
      "nested perfect necklaces",
      "normal numbers",
      "sobol-faure low discrepancy sequences",
      "real number",
      "pascal matrices modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}