{
  "abstract": "Let $\\mathscr L$ denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If $U\\subset\\mathbb R$ is an open interval and $f:U\\to\\mathbb R$ is real-analytic with $f(\\mathscr L\\cap U)\\subseteq\\mathscr L$, then $f$ is the restriction to $U$ of a rational function in $\\mathbb R(x)$. Quantitatively, there exists an absolute constant $τ>2$ such that, for every nonrational real-analytic function $f$ and every nonempty open subinterval $V\\subset U$, the set of $ξ\\in V\\cap\\mathscr L$ satisfying $μ(f(ξ))\\leqτ$ contains a Cantor set. As a consequence, every entire function $F:\\mathbb C\\to\\mathbb C$ satisfying $F(\\mathscr L)\\subseteq \\mathscr L$ is a polynomial with real coefficients. This gives a negative answer to a problem posed by Mahler in 1984. The proof develops a two-height counting estimate for rational approximation to analytic graphs, treating source and target denominators separately, and combines it with a nested construction producing Liouville inputs whose images have uniformly bounded irrationality exponent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14202",
  "author_total": 1,
  "authors": [
    {
      "name": "Diego Marques"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c72thptfgnrq4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14202",
  "fulltext_url": null,
  "id": "c72thptfgnrq4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14202",
  "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": "W7213254103",
  "pdf_url": "https://arxiv.org/pdf/2609.14202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c72thptfgnrq4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2609.14202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers",
  "title_text": "Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/c72thptfgnrq4",
  "venue": "arXiv",
  "versions": []
}