{
  "abstract": "Let $θ\\in\\mathbb Q\\setminus\\{0\\}$. We construct an approximating function $ψ:\\mathbb N\\to[0,\\frac12)$ for which \\[ \\sum_{q=1}^{\\infty}\\frac{φ(q)}{q}ψ(q)=\\infty, \\] but the set of $x\\in[0,1]$ for which \\[ |qx-a-θ|<ψ(q),\\qquad \\gcd(a,q)=1, \\] holds for infinitely many $(a,q)\\in\\mathbb Z\\times\\mathbb N$ has Lebesgue measure zero. Thus the inhomogeneous analogue of the Duffin--Schaeffer conjecture fails for every nonzero rational shift. By a modification of the construction, we further show that the set of shifts for which the inhomogeneous Duffin--Schaeffer conjecture fails is residual in $\\mathbb R$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30870",
  "author_total": 2,
  "authors": [
    {
      "name": "Yubin He"
    },
    {
      "name": "Lingmin Liao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0kpbwv0ss84yy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30870",
  "fulltext_url": null,
  "id": "0kpbwv0ss84yy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30870",
  "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": "W7214606883",
  "pdf_url": "https://arxiv.org/pdf/2609.30870",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0kpbwv0ss84yy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.30870",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30870"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts",
  "title_text": "Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/0kpbwv0ss84yy",
  "venue": "arXiv",
  "versions": []
}