{
  "abstract": "Bracketing covers and $δ$-covers provide finite discretizations of the anchored boxes that define the star discrepancy. Let $N_{[]}(d,δ)$ and $N(d,δ)$ denote the corresponding bracketing and covering numbers. We prove the lower bounds \\[ N_{[]}(d,δ)\\ge \\lceil δ^{-d}\\rceil, \\qquad N(d,δ)\\ge \\left\\lceil \\frac{d!}{d^d}\\,δ^{-d}\\right\\rceil. \\] We give two explicit constructions of bracketing covers. For every fixed $d$, together with the lower bound they imply $N_{[]}(d,δ)=(1+o_d(1))δ^{-d}$ as $δ\\downarrow0$. A first construction uses box-dependent anisotropic local grids and gives simple explicit bounds. A second, homothetic logarithmic-shell construction again attains this coefficient and gives $\\limsup_{d\\to\\infty}N_{[]}(d,δ)^{1/d}\\leδ^{-1}+e+O(δ)$ as $δ\\downarrow0$. Combining these finite estimates with Gnewuch's general bracketing bound and a Hoeffding--Bernstein chaining argument shows that, for every $d,n\\in\\mathbb N$, there exists an $n$-point set with star discrepancy at most $2.3463\\sqrt{d/n}$. Consequently, $\\lceil5.5052d\\varepsilon^{-2}\\rceil$ points suffice for star discrepancy at most $\\varepsilon$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00899",
  "author_total": 1,
  "authors": [
    {
      "name": "Kosuke Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ce7r2acrgy65s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00899",
  "fulltext_url": null,
  "id": "ce7r2acrgy65s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00899",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7172425325",
  "pdf_url": "https://arxiv.org/pdf/2608.00899",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ce7r2acrgy65s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2608.00899",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00899"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotically optimal bracketing covers for anchored boxes with applications to star discrepancy",
  "title_text": "Asymptotically optimal bracketing covers for anchored boxes with applications to star discrepancy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/ce7r2acrgy65s",
  "venue": "arXiv",
  "versions": []
}