{
  "abstract": "We study the generalized Flint Hills series $\\mathcal{F}_{s,t}(x)=\\sum_{n\\ge1} n^{-s}|\\sin(πnx)|^{-t}$ for $s>0$ and $t>1$. An explicit comparison with a series over continued-fraction denominators yields the Hausdorff dimension $\\min\\{1,2t/(s+t)\\}$ of its divergence set. At each critical exponent $τ=1+s/t>2$ we construct numbers of irrationality exponent $τ$ realizing both convergence and divergence; the convergent examples establish Meiburg's conjecture in the range $t>1$. Both parts of the critical fibre have Hausdorff dimension $2/τ$. For $s>t$ and $h_κ(r)=r^{2/τ}(\\log(1/r))^κ$ we prove that the divergence set has zero $h_κ$-measure for $κ<-1$ and infinite measure for $κ\\ge-1$. The divergent part of the critical fibre satisfies the same law, whereas its convergent part has infinite measure for every $κ$. The key estimate selects a rapidly growing subsequence of convergent denominators and gives a double-logarithmic bound on the approximation error. The convergence of the classical Flint Hills series $\\sum_{n\\ge1}(n^3\\sin^2 n)^{-1}$ remains undecided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12338",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuya Dan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bgftw4vj38wfp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12338",
  "fulltext_url": null,
  "id": "bgftw4vj38wfp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12338",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213008060",
  "pdf_url": "https://arxiv.org/pdf/2609.12338",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bgftw4vj38wfp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12338",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12338"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Critical convergence and Hausdorff measures for generalized Flint Hills series",
  "title_text": "Critical convergence and Hausdorff measures for generalized Flint Hills series",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/bgftw4vj38wfp",
  "venue": "arXiv",
  "versions": []
}