{
  "abstract": "Let $n$ be an integer with $n\\ge2$, and let $E$ be a real algebraic number field of degree $n+1$ over ${\\mathbb Q}$. Let $α_1, \\ldots , α_n$ be real numbers in $E$ such that $(1,α_1,\\ldots,α_n)$ is a linear basis of $E$ over ${\\mathbb Q}$. Let $ν_1, \\ldots, ν_{n-1}$ be real numbers satisfying $$0<\\max_{1\\le i\\le n-1}ν_i\\le2\\min_{1\\le i\\le n-1}ν_i,\\quad ν_1+ \\ldots +ν_{n-1}=1. $$ We establish that there exist a real number $C$, depending only on $α_1, \\ldots , α_n$, and infinitely many integers $Q \\ge 2$ satisfying the inequalities $$Q^{1/n}\\Vert Qα_i\\Vert \\le C (\\log Q)^{-ν_i}, \\quad 1\\le i\\le n-1, \\quad Q^{1/n}\\Vert Qα_n\\Vert\\le C.$$ This answers partially a conjecture of Peck, who proved in 1961 this statement in the particular case where $ν_1 = \\ldots = ν_{n-1} = 1/ (n-1)$. We also improve a result from 2004 of de Mathan and Teulié on a question of simultaneous Diophantine approximation involving a non-Archimedean valuation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29360",
  "author_total": 2,
  "authors": [
    {
      "name": "Yann Bugeaud"
    },
    {
      "name": "Bernard de Mathan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/azgv0dn10zw9a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29360",
  "fulltext_url": null,
  "id": "azgv0dn10zw9a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29360",
  "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": "W7214357266",
  "pdf_url": "https://arxiv.org/pdf/2609.29360",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/azgv0dn10zw9a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29360",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29360"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Refinements of Peck's theorem on simultaneous approximation to algebraic numbers",
  "title_text": "Refinements of Peck's theorem on simultaneous approximation to algebraic numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/azgv0dn10zw9a",
  "venue": "arXiv",
  "versions": []
}