{
  "abstract": "Schoof asked how many reduced Arakelov divisors cannot arise from the reduction algorithm. Tian and Tran subsequently characterized the image of this algorithm in real quadratic fields as precisely the $1$-reduced ideals. We give a quantitative answer to Schoof's question in the real-quadratic setting by determining the asymptotic proportion of reduced ideals missed by the algorithm. A key arithmetic input is the criterion that a reduced ideal with associated reduced form $(A,b,-r)$ fails to be $1$-reduced if and only if $A-r<b<2(A-r)$. Hence the missed ideals occupy an explicit region of the natural extension of the Gauss map whose relative measure is $λ_{\\mathrm{ex}}=\\frac{\\log(4/3)}{\\log4}$. In particular, the missed region occupies exactly half of the digit-one cylinder with respect to the invariant measure. We establish this limiting proportion in two settings. A lattice-point count gives the average asymptotic over fundamental discriminants $Δ\\le X$, with error $O(X^{-1/4+\\varepsilon})$. For individual real quadratic fields, Duke's equidistribution theorem yields $\\operatorname{ECal}(Δ)/κ(Δ) \\toλ_{\\mathrm{ex}}$ as $Δ\\to\\infty$. We also determine the limiting distribution of the canonical defect $k/\\sqrtΔ$ of the missed ideals. This distribution is supported on $[0,1/\\sqrt3-1/2]$, and its lower tail is asymptotic to $\\frac{2}{\\log(4/3)}t\\log(1/t)$ as $t\\to0^+$. An exact relation between the integer defect and an endpoint coordinate of the natural extension allows us to transfer equidistribution to the defect law.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36800",
  "author_total": 3,
  "authors": [
    {
      "name": "Mengzhe Guo"
    },
    {
      "name": "Taro Hayashi"
    },
    {
      "name": "Ryota Sueyosh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2pvb069rrhxrg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36800",
  "fulltext_url": null,
  "id": "2pvb069rrhxrg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36800",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7224100220",
  "pdf_url": "https://arxiv.org/pdf/2610.36800",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2pvb069rrhxrg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36800",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36800"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Density and Defect Distribution of Reduced Ideals Missed by the Reduction Algorithm in Real Quadratic Fields",
  "title_text": "Density and Defect Distribution of Reduced Ideals Missed by the Reduction Algorithm in Real Quadratic Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/2pvb069rrhxrg",
  "venue": "arXiv",
  "versions": []
}