{
  "abstract": "In the thirties of the last century, I.\\,M. Vinogradov proved that the inequality $\\|pα\\|\\le p^{-1/5+\\varepsilon}$ has infinitely many prime solutions $p$, where $\\|\\cdot\\|$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$, using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009), who showed that the inequality $\\|pα\\|\\le p^{-1/3+\\varepsilon}$ has infinitely many prime solutions. This exponent $1/3$ is considered the limit of current technology. Baier and the author established an analogue of Matomäki's result for the function field $k=\\Fq(T)$ and its imaginary quadratic extensions. Moreover, for real quadratic extensions of $k$ of class number one, they obtained an analogue of Vaughan's result (exponent $1/4$). In this paper, we establish the analogue of Matomäki's result for all real quadratic extensions of $k$ without any restriction on the class number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23742",
  "author_total": 1,
  "authors": [
    {
      "name": "Esrafil Ali Molla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4fhq56gpnp0ed.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23742",
  "fulltext_url": null,
  "id": "4fhq56gpnp0ed",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23742",
  "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": "W7222927706",
  "pdf_url": "https://arxiv.org/pdf/2610.23742",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4fhq56gpnp0ed.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23742",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23742"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The distribution of $αp$ modulo one in real quadratic function fields",
  "title_text": "The distribution of αp modulo one in real quadratic function fields",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/4fhq56gpnp0ed",
  "venue": "arXiv",
  "versions": []
}