Lévy-Khintchine Theorems: effective results and central limit theorems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911643577876480 |
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| author | Aggarwal, Gaurav Ghosh, Anish |
| author_facet | Aggarwal, Gaurav Ghosh, Anish |
| contents | 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 (\textit{Math. Annalen}, 1969) and Central Limit Theorems were proved by several authors \cites{Ibragimov, Misevicius, Morita, Vallee}. 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 (\textit{Annales scientifiques de l'ENS}, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20718 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lévy-Khintchine Theorems: effective results and central limit theorems Aggarwal, Gaurav Ghosh, Anish Number Theory Dynamical Systems Probability 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 (\textit{Math. Annalen}, 1969) and Central Limit Theorems were proved by several authors \cites{Ibragimov, Misevicius, Morita, Vallee}. 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 (\textit{Annales scientifiques de l'ENS}, 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. |
| title | Lévy-Khintchine Theorems: effective results and central limit theorems |
| topic | Number Theory Dynamical Systems Probability |
| url | https://arxiv.org/abs/2504.20718 |