{
  "id": "2012.14002",
  "version": "v1",
  "published": "2020-12-27T19:30:32.000Z",
  "updated": "2020-12-27T19:30:32.000Z",
  "title": "On the upper bound of the $L_2$-discrepancy of Halton's sequence",
  "authors": [
    "Mordechay B. Levin"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1806.11498",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $(H(n))_{n \\geq 0} $ be a $2-$dimensional Halton's sequence. Let $D_{2} ( (H(n))_{n=0}^{N-1}) $ be the $L_2$-discrepancy of $ (H_n)_{n=0}^{N-1} $. It is known that $\\limsup_{N \\to \\infty } (\\log N)^{-1} D_{2} ( H(n) )_{n=0}^{N-1} >0$. In this paper, we prove that $$D_{2} (( H(n) )_{n=0}^{N-1}) =O( \\log N) \\quad {\\rm for} \\; \\; N \\to \\infty ,$$ i.e., we found the smallest possible order of magnitude of $L_2$-discrepancy of a 2-dimensional Halton's sequence. The main tool is the theorem on linear forms in the $p$-adic logarithm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-27T19:30:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38"
    ],
    "keywords": [
      "upper bound",
      "discrepancy",
      "dimensional haltons sequence",
      "adic logarithm",
      "main tool"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}