{
  "id": "1501.02552",
  "version": "v1",
  "published": "2015-01-12T07:14:17.000Z",
  "updated": "2015-01-12T07:14:17.000Z",
  "title": "$L_p$-discrepancy of the symmetrized van der Corput sequence",
  "authors": [
    "Ralph Kritzinger",
    "Friedrich Pillichshammer"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "It is well known that the $L_p$-discrepancy for $p \\in [1,\\infty]$ of the van der Corput sequence is of exact order of magnitude $O((\\log N)/N)$. This however is for $p \\in (1,\\infty)$ not best possible with respect to the lower bounds according to Roth and Proinov. For the case $p=2$ it is well known that the symmetrization trick due to Davenport leads to the optimal $L_2$-discrepancy rate $O(\\sqrt{\\log N}/N)$ for the symmetrized van der Corput sequence. In this note we show that this result holds for all $p \\in (1,\\infty)$. The proof is based on an estimate of the Haar coefficients of the corresponding local discrepancy and on the use of the Littlewood-Paley inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-12T07:14:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38",
      "11K31"
    ],
    "keywords": [
      "symmetrized van der corput sequence",
      "exact order",
      "lower bounds",
      "symmetrization trick",
      "discrepancy rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}