{
  "abstract": "For a positive integer $n$, let $$S_0(n)=\\{1,2,\\ldots,\\lfloor n/2\\rfloor\\},\\qquad S_{j+1}(n)=\\{n\\bmod k:k\\in S_j(n)\\setminus\\{0\\}\\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \\cite{Ba-Vu}, Baraskar and Vukusic conjectured that for every fixed $j\\geq 2$, the limit $$\\lim_{n\\to\\infty} s_j(n)/n$$ does not exist. In this paper, we prove this conjecture in the affirmative. Moreover, we define a new class of iterated remainder sets $T_j(n)$ that naturally arises from the study of the length of the Engel series expansion of a rational number. We analogously study the asymptotic behavior of $t_j(n) := |T_j(n)|$. We show that $$\\lim_{n\\to\\infty}\\frac{t_j(n)}n$$ exists precisely for $j\\in\\{0,1\\}$ and fails to exist for every fixed integer $j\\geq2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24412",
  "author_total": 4,
  "authors": [
    {
      "name": "Omkar Baraskar"
    },
    {
      "name": "Prashant Gokhale"
    },
    {
      "name": "Sarvagya Jain"
    },
    {
      "name": "Adam Kieżun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5zy9jjm7z71e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24412",
  "fulltext_url": null,
  "id": "c5zy9jjm7z71e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24412",
  "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": "W7214048563",
  "pdf_url": "https://arxiv.org/pdf/2609.24412",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5zy9jjm7z71e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24412",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24412"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Asymptotic Behavior of Iterated Sets of Remainders",
  "title_text": "Asymptotic Behavior of Iterated Sets of Remainders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/c5zy9jjm7z71e",
  "venue": "arXiv",
  "versions": []
}