{
  "abstract": "Let $c>1$ be irrational. We prove that, for almost every $x>0$ and every $\\varepsilon>0$, there are infinitely many triples $(p,r,s)$, with $p$ and $r$ prime and $s$ an integer, for which \\[ 0<px-r<p^{-1/3+\\varepsilon},\\qquad 0<pcx-s<p^{-1/3+\\varepsilon}. \\] This improves the exponent $1/5$ in the homogeneous-line case of a theorem of Baier and Ghosh. The arithmetic input is the quantitative prime-counting lower bound in Matomäki's proof concerning primes in shrinking neighborhoods of an irrational rotation. We obtain a local first-moment lower bound by fixing the prime numerator and applying the prime number theorem to the admissible prime denominators. Two Selberg sieve squares, together with explicit trigonometric interval approximations, give a pointwise upper bound whose error tends to zero in $L^1$ after normalization. A truncation and density-point argument then yields the almost-everywhere conclusion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36380",
  "author_total": 1,
  "authors": [
    {
      "name": "Zirui Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4b5hg7eek42ck.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36380",
  "fulltext_url": null,
  "id": "4b5hg7eek42ck",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36380",
  "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": "W7224128345",
  "pdf_url": "https://arxiv.org/pdf/2610.36380",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4b5hg7eek42ck.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36380",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36380"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Metric Diophantine approximation with primes on irrational lines",
  "title_text": "Metric Diophantine approximation with primes on irrational lines",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/4b5hg7eek42ck",
  "venue": "arXiv",
  "versions": []
}