{
  "abstract": "We prove two results on the fine structure of the binary digits of $3^{m}$. First, for every fixed period $p$, the number of positions at which the binary expansion of $3^{m}$ breaks $p$-periodicity grows in order like $\\log m/\\log\\log m$; equivalently, no window of the expansion deeper than a fixed power of $\\log m$ is $p$-periodic. Second, the finite binary word formed by the low-order digits of $3^{m}$ has full low-order subword complexity: its complexity function satisfies $p_{3^{m}}(n)\\ge n+1$ for every length $n$, once $m$ is large enough.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14774",
  "author_total": 1,
  "authors": [
    {
      "name": "Ralf Stephan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/854fyytf9cgab.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14774",
  "fulltext_url": null,
  "id": "854fyytf9cgab",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14774",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7169558220",
  "pdf_url": "https://arxiv.org/pdf/2607.14774",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/854fyytf9cgab.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.14774",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14774"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Aperiodicity and subword complexity in the binary expansion of powers of three",
  "title_text": "Aperiodicity and subword complexity in the binary expansion of powers of three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/854fyytf9cgab",
  "venue": "arXiv",
  "versions": []
}