{
  "id": "1601.07281",
  "version": "v1",
  "published": "2016-01-27T07:42:55.000Z",
  "updated": "2016-01-27T07:42:55.000Z",
  "title": "$L_p$-discrepancy and beyond of higher order digital sequences",
  "authors": [
    "Josef Dick",
    "Aicke Hinrichs",
    "Lev Markhasin",
    "Friedrich Pillichshammer"
  ],
  "categories": [
    "math.NT",
    "math.FA",
    "math.NA"
  ],
  "abstract": "The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution modulo one of infinite sequences. In 1986 Proinov proved for all $p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences in the $d$-dimensional unit cube, but it remained an open question whether this lower bound is best possible in the order of magnitude until recently. In 2014 Dick and Pillichshammer gave a first construction of an infinite sequence whose order of $L_2$-discrepancy matches the lower bound of Proinov. Here we give a complete solution to this problem for all finite $p > 1$. We consider so-called order $2$ digital $(t,d)$-sequences over the finite field with two elements and show that such sequences achieve the optimal order of $L_p$-discrepancy simultaneously for all $p \\in (1,\\infty)$. Beyond this result, we estimate the norm of the discrepancy function of those sequences also in the space of bounded mean oscillation, exponential Orlicz spaces, Besov and Triebel-Lizorkin spaces and give some corresponding lower bounds which show that the obtained upper bounds are optimal in the order of magnitude.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-27T07:42:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K06",
      "11K38",
      "32A37",
      "42C10",
      "46E30",
      "46E35",
      "65C05"
    ],
    "keywords": [
      "higher order digital sequences",
      "discrepancy",
      "lower bound",
      "general infinite sequences",
      "dimensional unit cube"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160107281D"
    }
  }
}