{
  "abstract": "Consider the random lacunary series $f(z) = \\sum_{k=1}^\\infty \\frac{ξ_k}{\\sqrt{k}} \\, z^{2^k}$ on the unit disk, where $\\{ ξ_k \\}$ are independent standard complex Gaussian random variables. We show that almost surely, a.e. $ζ\\in \\partial \\mathbb{D}$ is a Plessner point of $f$, yet the image of every Stolz angle with vertex at $ζ$ has asymptotic density zero. This gives a negative answer to questions of Collingwood and Baernstein concerning possible strengthenings of Plessner's theorem. In this example, for a.e. $ζ\\in \\partial \\mathbb{D}$, the non-tangential range of $f$ at $ζ$ has zero area. The non-tangential range cannot be much smaller: we show that for an arbitrary holomorphic function on the unit disk, the non-tangential range has Hausdorff dimension 2 at almost every Plessner point.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18785",
  "author_total": 1,
  "authors": [
    {
      "name": "Oleg Ivrii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d5ccgfnkvbgm8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18785",
  "fulltext_url": null,
  "id": "d5ccgfnkvbgm8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18785",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d5ccgfnkvbgm8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Non-tangential ranges of holomorphic functions at Plessner points",
  "title_text": "Non-tangential ranges of holomorphic functions at Plessner points",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/d5ccgfnkvbgm8",
  "venue": "arXiv",
  "versions": []
}