{
  "abstract": "For a compact set $K\\subset\\mathbb{C}$, let $\\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros. We prove that $\\vartheta(K)=0$ whenever $\\operatorname{cap}(K)=1$, with no regularity assumption on $K$. The proof uses a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of $K$. A Fourier average of exterior harmonic measures realizes this polynomial as the logarithmic potential of a signed measure having bounded density with respect to the equilibrium measure. A positive perturbation and an $L^1$ approximation by empirical measures then produce the required polynomials. This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran to arbitrary compact sets of capacity one. Together with the capacity-greater-than-one theorem of Ghosh and Ramachandran and an elementary argument for unbounded sets, it follows that $\\vartheta(F)=0$ for every closed infinite set $F\\subset\\mathbb{C}$ of transfinite diameter at least one, answering the vanishing question in Erdős Problem 1040.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06050",
  "author_total": 1,
  "authors": [
    {
      "name": "Ioannis Tzachristas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/frrnjh6y8jgn1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06050",
  "fulltext_url": null,
  "id": "frrnjh6y8jgn1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06050",
  "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",
      "cs.AI"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7212021770",
  "pdf_url": "https://arxiv.org/pdf/2609.06050",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/frrnjh6y8jgn1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06050",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06050"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A solution to the Erdős Problem #1040",
  "title_text": "A solution to the Erdős Problem #1040",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/frrnjh6y8jgn1",
  "venue": "arXiv",
  "versions": []
}