{
  "abstract": "We study quantitative density of orbits of the free semigroup $Γ$ generated by two maps of $(0,\\infty)$. For the Möbius pair $x\\mapsto 1/x$, $x\\mapsto x+1$ we show that every orbit is $O(n^{-1})$-dense in $[0,\\infty)$, that every irrational number $u$ is approached at a rate governed by the continued fraction of $u$ (at least $n^{-2}$, exponential when $u$ has bounded partial quotients, and $\\exp(-cn/\\log n)$ for almost every $u$), and that for rational $u$ and rational starting points the rate $n^{-1}$ is optimal. For the affine pair $T_0(x)=x/a$, $T_1(x)=bx+1$ with $a,b>1$, which was studied by Bergelson, Misiurewicz and Senti, we prove that for every $x>0$ and $u\\ge 0$ there are infinitely many $γ\\inΓ$ with \\[ |γ(x)-u|<C\\,λ^{-|γ|},\\qquad λ=\\exp\\!\\Big(\\frac{\\log a\\,\\log\\frac{a}{a-1}}{\\log\\frac{a^2b}{a-1}}\\Big)>1, \\] where $|γ|$ is the word length of $γ$ and $C$ is independent of $γ$. This exponential rate holds whether or not $\\log_a b$ is rational, and the exponential order cannot be improved for almost every $u$. We also give a corrected version of the Kronecker-type estimate $|γ(x)-u|<(u+1)(t^{1/|γ|}-1)$. Finally, we discuss the approximation of real numbers by ratios of primes and by ratios of logarithms of primes, updated in light of recent results on gaps between primes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: M. Javaheri, A generalization of Dirichlet approximation theorem for the affine actions on real line, J. Number Theory 128 (2008), no. 5, 1146--1156",
  "arxiv": "2611.03778",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammad Javaheri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bpxq4z8qjxrym.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03778",
  "fulltext_url": null,
  "id": "bpxq4z8qjxrym",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03778",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03778",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bpxq4z8qjxrym.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03778",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03778"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A generalization of the Dirichlet approximation theorem for affine actions on the real line",
  "title_text": "A generalization of the Dirichlet approximation theorem for affine actions on the real line",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/bpxq4z8qjxrym",
  "venue": "arXiv",
  "versions": []
}