{
  "id": "1505.04975",
  "version": "v1",
  "published": "2015-05-19T12:53:09.000Z",
  "updated": "2015-05-19T12:53:09.000Z",
  "title": "On the lower bound of the discrepancy of $(t,s)$ sequences: II",
  "authors": [
    "Mordechay B. Levin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $ (\\bx(n))_{n \\geq 1} $ be an $s-$dimensional Niederreiter-Xing sequence in base $b$. Let $D((\\bx(n))_{n = 1}^{N})$ be the discrepancy of the sequence $ (\\bx(n))_{n = 1}^{N} $. It is known that $N D((\\bx(n))_{n = 1}^{N}) =O(\\ln^s N)$ as $N \\to \\infty $. In this paper, we prove that this estimate is exact. Namely, there exists a constant $K>0$, such that $$ \\inf_{\\bw \\in [0,1)^s} \\sup_{1 \\leq N \\leq b^m} N D((\\bx(n)\\oplus \\bw)_{n = 1}^{N}) \\geq K m^s \\quad {\\rm for} \\; \\; m=1,2,...\\;. $$ We also get similar results for other explicit constructions of $(t,s)$ sequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-19T12:53:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38"
    ],
    "keywords": [
      "lower bound",
      "discrepancy",
      "dimensional niederreiter-xing sequence",
      "similar results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150504975L"
    }
  }
}