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<p>The Lévy–Khintchine theorem is a classical result in Diophantine approximation that describes the asymptotic growth of the denominators of convergents in the continued fraction expansion of a typical real number. An effective version of this theorem was proved by Phillip and Stackelberg (Math Annalen, 1969) and Central Limit Theorems were proved by several authors (Ibragimov in Vestnik Leningrad Univ 16:13–24, 1961, Misevičius in Lith Math J 21:245–253, 1981. <a href="https://doi.org/10.1007/BF01116883">https://doi.org/10.1007/BF01116883</a>, Morita in J Math Soc Jpn 46:309–343, 1994. <a href="https://doi.org/10.2969/jmsj/04620309">https://doi.org/10.2969/jmsj/04620309</a>, Vallée in Acta Arith 81:101–144, 1997. <a href="https://doi.org/10.4064/aa-81-2-101-144">https://doi.org/10.4064/aa-81-2-101-144</a>). In this work, we develop a new approach towards quantifying the Lévy–Khintchine theorem. Our methods apply to the setting of higher-dimensional simultaneous Diophantine approximation, thereby providing an effective version of a theorem of Cheung and Chevallier (Ann Sci Éc Norm Supér (4) 57:185–240, 2024). Further, we prove a Central Limit Theorem for best approximations in all dimensions. Unlike previous approaches to the one-dimensional problem, our approach relies on techniques from homogeneous dynamics.</p>

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Lévy–Khintchine theorems: effective results and central limit theorems

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Gaurav Aggarwal,
  • Anish Ghosh

摘要

The Lévy–Khintchine theorem is a classical result in Diophantine approximation that describes the asymptotic growth of the denominators of convergents in the continued fraction expansion of a typical real number. An effective version of this theorem was proved by Phillip and Stackelberg (Math Annalen, 1969) and Central Limit Theorems were proved by several authors (Ibragimov in Vestnik Leningrad Univ 16:13–24, 1961, Misevičius in Lith Math J 21:245–253, 1981. https://doi.org/10.1007/BF01116883, Morita in J Math Soc Jpn 46:309–343, 1994. https://doi.org/10.2969/jmsj/04620309, Vallée in Acta Arith 81:101–144, 1997. https://doi.org/10.4064/aa-81-2-101-144). In this work, we develop a new approach towards quantifying the Lévy–Khintchine theorem. Our methods apply to the setting of higher-dimensional simultaneous Diophantine approximation, thereby providing an effective version of a theorem of Cheung and Chevallier (Ann Sci Éc Norm Supér (4) 57:185–240, 2024). Further, we prove a Central Limit Theorem for best approximations in all dimensions. Unlike previous approaches to the one-dimensional problem, our approach relies on techniques from homogeneous dynamics.