{
  "abstract": "For $n$-dimensional simultaneous best Diophantine approximations in an arbitrary norm induced by an inner product, for $n\\geq2$ we prove $q_{k+2^n}\\geq q_k+\\min\\{q_{k+2^{n-1}},2q_{k+1}\\}$. This yields $g_n(α):=\\liminf_{m\\to\\infty}(q_m)^{1/m}\\geqφ^{1/2^{n-1}}, \\text{ for } \\ φ= \\dfrac{1 + \\sqrt{5}}{2}.$ Consequently, $\\displaystyle G(n):=\\inf_{α\\in\\mathbb R^n\\setminus\\mathbb Q^n}g_n(α)\\geqφ^{1/2^{n-1}}$ and $\\underline{\\mathcal D}_n(α)\\leq\\left\\lfloor 2^{n-1}\\frac{\\log2}{\\logφ}\\right\\rfloor+1$, where $\\underline{\\mathcal D}_n(α)$ is a quantity related to multidimensional generalizations of the three-distance theorem. In particular, $g_2(α)\\geq\\sqrtφ, \\, g_3(α)\\geq\\sqrt[4]φ, \\, \\underline{\\mathcal D}_2(α)\\leq3, \\, \\underline{\\mathcal D}_3(α)\\leq6.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02386",
  "author_total": 1,
  "authors": [
    {
      "name": "Leonid M. Shatunov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tyme5r89x94p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02386",
  "fulltext_url": null,
  "id": "9tyme5r89x94p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02386",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7207838037",
  "pdf_url": "https://arxiv.org/pdf/2609.02386",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tyme5r89x94p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.02386",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02386"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Growth of Denominators of Simultaneous Best Diophantine Approximations in a Norm Induced by an Inner Product",
  "title_text": "On the Growth of Denominators of Simultaneous Best Diophantine Approximations in a Norm Induced by an Inner Product",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9tyme5r89x94p",
  "venue": "arXiv",
  "versions": []
}