{
  "abstract": "For integers $1\\leq a\\leq b$, the length of the finite Pierce expansion of $a/b$ is the number of steps of the iteration $r\\mapsto b\\bmod r$ which starts at $a$ and ends at zero. Erdős and Shallit (1991) proved that the average of these lengths over $1\\leq a\\leq b$ is bounded below by a positive constant times $\\log\\log b$, and suggested that it is at least of order $\\log b$. In this paper, we prove that the average length is at least $\\sqrt{2\\log b}-3(\\log b)^{1/4}-11$ for every $b\\geq2$. We also show that, for each $0<c<\\sqrt2$, the proportion of numerators $a$ for which the length is less than $c\\sqrt{\\log b}$ tends to zero exponentially fast in $\\sqrt{\\log b}$, with an explicit rate. Our approach refines the interval-counting argument of Erdős and Shallit by restricting the product of the initial digits. As applications, we obtain similar bounds for reduced fractions and for numerators in a fixed congruence class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26594",
  "author_total": 1,
  "authors": [
    {
      "name": "Min Woong Ahn"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8a6qpj1b1q1hw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26594",
  "fulltext_url": null,
  "id": "8a6qpj1b1q1hw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26594",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223044198",
  "pdf_url": "https://arxiv.org/pdf/2610.26594",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8a6qpj1b1q1hw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26594",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26594"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Uniform lower bounds for the average length of finite Pierce expansions",
  "title_text": "Uniform lower bounds for the average length of finite Pierce expansions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/8a6qpj1b1q1hw",
  "venue": "arXiv",
  "versions": []
}