{
  "abstract": "Let $x_{0}$ be a positive integer, let $x_{n}=\\lceil 3x_{n-1}/2\\rceil$, and let $w_{n}=2x_{n+1}-3x_{n}\\in\\{0,1\\}$ be the associated word studied by Dubickas; for $x_{0}=1$ the orbit is A061419 and $w$ is the minimal word $g_{3/2}$ of the rational base number system of Akiyama, Frougny and Sakarovitch. The orbit encodes a real constant $K=\\lim_{n}x_{n}(2/3)^{n}$, equal for $x_{0}=1$ to $ω_{3/2}=K(3)=1.6222705028\\ldots$, whose irrationality has been open since 1977. We prove that if $w$ is automatic then $K$ is transcendental; equivalently, an algebraic $K$ forces $w$ to be non-automatic. Further we prove that either the complexity of $w$ exceeds every linear bound or $K$ is irrational. All results are formally verified in the Lean~4 proof assistant, on three cited axioms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19007",
  "author_total": 1,
  "authors": [
    {
      "name": "Ralf Stephan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bz0pxh0da8fs1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19007",
  "fulltext_url": null,
  "id": "bz0pxh0da8fs1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19007",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213498656",
  "pdf_url": "https://arxiv.org/pdf/2609.19007",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bz0pxh0da8fs1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19007",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19007"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Transcendence criteria for the minimal word of the rational base $3/2$",
  "title_text": "Transcendence criteria for the minimal word of the rational base 3/2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/bz0pxh0da8fs1",
  "venue": "arXiv",
  "versions": []
}