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<p>We prove that for any <i>η</i> that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets <i>k</i><sup>−1</sup>((−∞, <i>η</i>]) and <i>k</i><sup>−1</sup>(<i>η</i>), which are the sets of irrational numbers with best constant of Diophantine approximation bounded by <i>η</i> and exactly <i>η</i> respectively, have the same Hausdorff dimension. We also show that, as <i>η</i> varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.</p>

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Concentration of Dimension in Extremal Points of Left-half Lines in the Lagrange Spectrum

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Carlos Gustavo Moreira,
  • Christian Camilo Silva Villamil

摘要

We prove that for any η that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets k−1((−∞, η]) and k−1(η), which are the sets of irrational numbers with best constant of Diophantine approximation bounded by η and exactly η respectively, have the same Hausdorff dimension. We also show that, as η varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.