{
  "abstract": "For a positive integer $n$, let $$ S_1(n)=\\{n\\bmod k:1\\leq k\\leq\\lfloor n/2\\rfloor\\}, \\qquad s_1(n)=|S_1(n)|. $$ Baraskar and Vukusic proved $s_1(n)=Ξn+O(n/(\\log n\\log\\log n))$ for an explicit positive constant $Ξ$, and asked for a sharper error term. Their computations suggested that an $O(n^{1/3})$ bound might hold. We prove that, for every $α>\\sqrt2$, $$ \\limsup_{n\\to\\infty} \\frac{s_1(n)-Ξn}{n\\exp\\{-α\\sqrt{\\log n\\log\\log n}\\}} =\\infty, $$ and the corresponding limit inferior is $-\\infty$. In particular, no fixed power saving is possible. We also prove the complementary upper estimate $$ s_1(n)=Ξn+O\\left( n\\exp\\{-α\\sqrt{\\log n\\log\\log n}\\} \\right) $$ for every fixed $α\\in (0,\\sqrt{2})$, sharpening the previous $O(n/(\\log n\\log\\log n))$ bound. Thus, the constant $\\sqrt2$ is the threshold for bounds of the form $n\\exp\\{-α\\sqrt{\\log n\\log\\log n}\\}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04993",
  "author_total": 4,
  "authors": [
    {
      "name": "Omkar Baraskar"
    },
    {
      "name": "Prashant Gokhale"
    },
    {
      "name": "Sarvagya Jain"
    },
    {
      "name": "Adam Kieżun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80pmcex1bprnz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04993",
  "fulltext_url": null,
  "id": "80pmcex1bprnz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04993",
  "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": "W7220704611",
  "pdf_url": "https://arxiv.org/pdf/2610.04993",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80pmcex1bprnz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04993",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04993"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Error Term Estimates for Sets of Remainders",
  "title_text": "Error Term Estimates for Sets of Remainders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/80pmcex1bprnz",
  "venue": "arXiv",
  "versions": []
}