{
  "abstract": "We develop a new framework, inspired by Schmidt's games, to study the Lagrange spectrum for simultaneous Diophantine approximation in dimension $d\\geq 2$. We show the Hausdorff dimension of the set of points in $\\mathbb{R}^{d}$ whose best approximation constant lies in $[\\varepsilon,\\varepsilon(1+δ\\varepsilon^{d})]$, for some constant $δ>0$, is positive, and for a slightly larger set approaches full dimension as $\\varepsilon \\to 0$. The proof combines a novel application of the Simplex lemma near rational points with the game-theoretic framework. We also use an elementary observation to show that the naturally defined Lagrange spectrum for systems of linear forms is uncountable in the case of square matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05311",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Ward"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chd2jkjwy25b0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05311",
  "fulltext_url": null,
  "id": "chd2jkjwy25b0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05311",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7211915264",
  "pdf_url": "https://arxiv.org/pdf/2609.05311",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chd2jkjwy25b0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.05311",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05311"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Distribution of points near the origin in the $d$-dimensional Lagrange spectrum",
  "title_text": "Distribution of points near the origin in the d-dimensional Lagrange spectrum",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/chd2jkjwy25b0",
  "venue": "arXiv",
  "versions": []
}