{
  "abstract": "Let $α\\in \\mathbb{R} \\setminus \\mathbb{Q}$, $β\\in \\R$, $N \\in \\mathbb{R}_{\\ge 1}$ and $ Δ\\in (0, 1/2)$. For any real $y$, let $\\|y\\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \\ge 1$, there are infinitely many primes $\\ell \\equiv u \\bmod v$ such that $ \\|α\\ell - β\\| \\ll_v \\ell^{-1/4} \\log^{8} \\ell. $ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary. Before stating the result, let us define a function on $f_Δ(θ)$ on $\\R$ such that $f_Δ(θ)$ is $ 1 \\text{ if } \\| θ\\| < Δ$ and $ 0 $ otherwise. Further, suppose that $u, v \\in \\mathbb{Z}_{\\ge 1}$ are coprime and $a$, $q \\in \\Z$ are coprime with $q> N^{1/4}$ and $|αv -a/q| \\le 1/q^2$. Then, for every $ε\\in \\mathbb{R}_{>0}$ we have \\begin{equation*} \\sum_{\\substack{n=1 \\\\ n \\equiv u \\bmod v}}^N Λ(n) (f_Δ(αn - β) - 2Δ) \\ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}Δ^{1/2} + (ΔNq)^{1/2} + N^εq Δ^{1-ε}) \\mathcal{L}^8 \\end{equation*} where $\\mathcal{L}=\\log (Nq/Δ)$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \\begin{equation*} \\sum_{\\substack{n=1}}^N Λ(n) (f_Δ(αn - β) - 2Δ). \\end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \\subset \\R\\setminus \\Q$ such that for every $γ\\in S$ there are infinitely many primes $\\ell \\equiv u \\bmod v$ satisfying $\\| γ\\ell \\| < \\ell^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erd{ö}s and Mahler from 1939.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10010",
  "author_total": 2,
  "authors": [
    {
      "name": "D. Mazumder"
    },
    {
      "name": "J. Sivaraman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7z9q7yjkqy2ge.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10010",
  "fulltext_url": null,
  "id": "7z9q7yjkqy2ge",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10010",
  "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": "W7212119502",
  "pdf_url": "https://arxiv.org/pdf/2609.10010",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7z9q7yjkqy2ge.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10010",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10010"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Diophantine approximation with primes in an arithmetic progression",
  "title_text": "Diophantine approximation with primes in an arithmetic progression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/7z9q7yjkqy2ge",
  "venue": "arXiv",
  "versions": []
}