{
  "id": "2109.06461",
  "version": "v1",
  "published": "2021-09-14T06:11:07.000Z",
  "updated": "2021-09-14T06:11:07.000Z",
  "title": "Exact order of extreme $L_p$ discrepancy of infinite sequences in arbitrary dimension",
  "authors": [
    "Ralph Kritzinger",
    "Friedrich Pillichshammer"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the extreme $L_p$ discrepancy of infinite sequences in the $d$-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star $L_p$ discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension $d$ and any $p>1$ the extreme $L_p$ discrepancy of every infinite sequence in $[0,1)^d$ is at least of order of magnitude $(\\log N)^{d/2}$, where $N$ is the number of considered initial terms of the sequence. For $p \\in (1,\\infty)$ this order of magnitude is best possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-14T06:11:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38",
      "11K06",
      "11K31"
    ],
    "keywords": [
      "infinite sequence",
      "arbitrary dimension",
      "exact order",
      "discrepancy",
      "test sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}