{
  "abstract": "We develop a general method for uniform twisted metric Diophantine approximation with an arbitrary fixed real matrix. Using all successive minima of the associated diagonal lattice trajectory, we estimate lattice-point counts, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted $ψ$-Dirichlet vectors for a large class of functions $ψ$, including all except finitely many power functions {$T^{-τ}$, $τ>0$}. We obtain Hausdorff dimension formulae for the endpoint sets obtained by taking the intersection and union of the sets of twisted $cψ$-Dirichlet vectors over $c>0$, and prove that taking the intersections causes no dimension drop. Furthermore, we obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent. In addition, we prove zero-full laws for general Hausdorff measures of $ψ$-Dirichlet sets. Applications recover the one-dimensional Kim-Liao formulae, strengthen Aggarwal's escape-of-mass upper bound, reproduce the metric criteria of Moshchevitin and the second author and give certain extensions of the above statements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36412",
  "author_total": 2,
  "authors": [
    {
      "name": "Taehyeong Kim"
    },
    {
      "name": "Vasiliy Neckrasov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aj27v4jvpcms2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36412",
  "fulltext_url": null,
  "id": "aj27v4jvpcms2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36412",
  "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": "W7214887149",
  "pdf_url": "https://arxiv.org/pdf/2609.36412",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aj27v4jvpcms2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36412",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36412"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Twisted Diophantine Approximation II: Uniform Theory",
  "title_text": "Twisted Diophantine Approximation II: Uniform Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/aj27v4jvpcms2",
  "venue": "arXiv",
  "versions": []
}