{
  "abstract": "For an integer $k\\geq2$, let $α_k$ be the supremum of the real numbers $a$ for which almost every integer $n\\geq2$ has divisors $d_1<\\cdots<d_k\\mid n$ satisfying $d_k\\leq d_1\\bigl(1+(\\log n)^{-a}\\bigr).$ Let $\\mathcal A\\subseteq\\N$ be the logarithmic random set in which the events $m\\in\\mathcal A$ are mutually independent and $\\Pp(m\\in\\mathcal A)=1/m$ for every $m\\geq1$. For a finite set $B\\subseteq\\N$, write $Σ(B)=\\sum_{b\\in B}b, Σ(\\varnothing)=0$ and $m(B)=\\max_{s\\in\\Z}\\#\\{C\\subseteq B\\midΣ(C)=s\\},$ and define \\[ β_k=\\sup\\left\\{c<1\\,\\middle|\\,\\lim_{D\\to\\infty}\\Pp\\bigl(m(\\mathcal A\\cap(D^c,D])\\geq k\\bigr)=1\\right\\}. \\] Ford, Green and Koukoulopoulos proved $α_k\\geqβ_k/(1-β_k)$ and conjectured that equality holds for every fixed $k\\geq2$. In this paper, we confirm their conjecture. More precisely, for every fixed $a>β_k/(1-β_k)$, almost every integer $n\\geq2$ has no divisors $d_1<\\cdots<d_k\\mid n$ satisfying $d_k\\leq d_1\\bigl(1+(\\log n)^{-a}\\bigr)$. We also correct local errors in their paper [\\emph{Invent. Math.} 232 (2023), 1027--1160], concerning the finite-subflag reduction, the residual-sum count, the moment estimate and the lattice adjustment. These corrections preserve the entropy-threshold comparison used in our proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22296",
  "author_total": 2,
  "authors": [
    {
      "name": "Yaping Mao"
    },
    {
      "name": "Yanyan Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zxmhdmrsgt8y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22296",
  "fulltext_url": null,
  "id": "7zxmhdmrsgt8y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22296",
  "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.CO"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214057449",
  "pdf_url": "https://arxiv.org/pdf/2609.22296",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zxmhdmrsgt8y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.22296",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22296"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture",
  "title_text": "Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/7zxmhdmrsgt8y",
  "venue": "arXiv",
  "versions": []
}