{
  "abstract": "We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point counts. An equivalent formulation is using all successive minima of the associated diagonal lattice trajectory, capturing homogeneous clustering in every direction. We also obtain an almost-sure zero-infinity law for inhomogeneous $ψ$-Lagrange constants, separate Hausdorff-measure criteria, and dimension results. Within this regularity class, our theorem recovers the Kurzweil and Fuchs-Kim criteria. At the critical exponent, this series determines whether the set of badly approximable shifts has zero or full Lebesgue measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36411",
  "author_total": 2,
  "authors": [
    {
      "name": "Taehyeong Kim"
    },
    {
      "name": "Vasiliy Neckrasov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa82532r8g748.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36411",
  "fulltext_url": null,
  "id": "fa82532r8g748",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36411",
  "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",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214949361",
  "pdf_url": "https://arxiv.org/pdf/2609.36411",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa82532r8g748.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36411",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36411"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Twisted Diophantine Approximation I: Asymptotic Theory",
  "title_text": "Twisted Diophantine Approximation I: Asymptotic Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/fa82532r8g748",
  "venue": "arXiv",
  "versions": []
}