{
  "abstract": "We study the one-dimensional Dirichlet spectrum and obtain new results concerning the origin $μ$ of the ray contained in the spectrum and its isolated points. First, we prove the improved upper bound $μ\\leq (448875+1611\\sqrt{67773})/46982 = 18.4809\\ldots$. We also exhibit an explicit gap $(L,R)$ in the Dirichlet spectrum, which yields the improved lower bound $μ\\geq R = 5.9596\\ldots$. The proof of the lower bound includes an exact computer-assisted finite search over admissible continued-fraction words. We further prove that all points $Z_k$, $k\\geq 2$, introduced by Ivanov are isolated points of the Dirichlet spectrum. Finally, we determine the asymptotic behavior of the neighboring gaps around $Z_k$ and describe the equivalence classes of all irrational numbers whose Dirichlet value is $Z_k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03787",
  "author_total": 1,
  "authors": [
    {
      "name": "Maksim Lavrov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6zb8rfgkh3sj2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03787",
  "fulltext_url": null,
  "id": "6zb8rfgkh3sj2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03787",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220706998",
  "pdf_url": "https://arxiv.org/pdf/2610.03787",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6zb8rfgkh3sj2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03787",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03787"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum",
  "title_text": "Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/6zb8rfgkh3sj2",
  "venue": "arXiv",
  "versions": []
}