{
  "id": "2210.08441",
  "version": "v1",
  "published": "2022-10-16T04:13:16.000Z",
  "updated": "2022-10-16T04:13:16.000Z",
  "title": "On the one-sided boundedness of the local discrepancy of $\\{nα\\}$-sequences",
  "authors": [
    "Jiangang Ying",
    "Yushu Zheng"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The main interest of this article is the one-sided boundedness of the local discrepancy of $\\alpha\\in\\mathbb{R}\\setminus\\mathbb{Q}$ on the interval $(0,c)\\subset(0,1)$ defined by \\[D_n(\\alpha,c)=\\sum_{j=1}^n 1_{\\{\\{j\\alpha\\}<c\\}}-cn.\\] We focus on the special case $c\\in (0,1)\\cap\\mathbb{Q}$. Several necessary and sufficient conditions on $\\alpha$ for $(D_n(\\alpha,c))$ to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set \\[O_c=\\{\\alpha\\in \\irr: (D_n(\\alpha,c)) \\text{ is one-side bounded}\\}.\\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-16T04:13:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K06",
      "11K50"
    ],
    "keywords": [
      "local discrepancy",
      "one-sided boundedness",
      "main interest",
      "special case",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}