{
  "abstract": "In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. In 1977, Nagasaka asked whether this exceptional set must have logarithmic capacity zero. We prove that there exist a transcendental entire function and a regular local inverse for which the exceptional set contains a compact set of positive logarithmic capacity. In fact, for every $0<s<1$, such a function and inverse can be chosen so that this compact set has positive $s$-dimensional Hausdorff measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29797",
  "author_total": 1,
  "authors": [
    {
      "name": "Sina Nadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8abdkcq0rcnzc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29797",
  "fulltext_url": null,
  "id": "8abdkcq0rcnzc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29797",
  "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.CV"
    ],
    [
      "openalex",
      "2604|Meromorphic and Entire Functions"
    ]
  ],
  "openalex": "W7214293107",
  "pdf_url": "https://arxiv.org/pdf/2609.29797",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8abdkcq0rcnzc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29797",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29797"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Exceptional Sets of Positive Capacity in Gross's Star Theorem",
  "title_text": "Exceptional Sets of Positive Capacity in Gross's Star Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/8abdkcq0rcnzc",
  "venue": "arXiv",
  "versions": []
}