{
  "abstract": "Let $ξ$ be a real algebraic irrational number and $p(n,ξ,b)$ the number of distinct blocks of length $n$ in its base-$b$ expansion. Bugeaud and Evertse proved in 2008 that $p(n,ξ,b)/(n(\\log n)^η)$ is unbounded for every $η<1/11$. We prove this for every $η<1/4$. We also prove the analogous bound for the continued fraction expansion of an algebraic number of degree at least three, with every exponent $δ<1/4$. Bugeaud announced this statement in 2013 with an unspecified $δ>0$. Finally, the subword complexity of the steering word of $(3/2)^n$ exceeds $C\\,k(\\log k)^η$ infinitely often for every $η<1/4$ and every $C$. The three proofs share one count: $\\varepsilon^{-1}$ classes times $\\varepsilon^{-3}$ subspaces from the quantitative Subspace Theorem of Evertse and Ferretti. The subspaces are then disposed of at the cost of a logarithm, with no further power of $\\varepsilon^{-1}$. All three results, and the quantitative Subspace Theorem they rest on, are machine-verified in Lean~4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30937",
  "author_total": 1,
  "authors": [
    {
      "name": "Ralf Stephan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cqg1fkfsb1rt8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30937",
  "fulltext_url": null,
  "id": "cqg1fkfsb1rt8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30937",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223279033",
  "pdf_url": "https://arxiv.org/pdf/2610.30937",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cqg1fkfsb1rt8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30937",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30937"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The exponent $1/4$ in three complexity bounds: integer bases, continued fractions, and the $(3/2)^n$ steering word",
  "title_text": "The exponent 1/4 in three complexity bounds: integer bases, continued fractions, and the (3/2)^n steering word",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/cqg1fkfsb1rt8",
  "venue": "arXiv",
  "versions": []
}