Generalized Lévy-Khintchine Theorems and a Conjecture of Y. Cheung
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908474184564736 |
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| author | Aggarwal, Gaurav Ghosh, Anish |
| author_facet | Aggarwal, Gaurav Ghosh, Anish |
| contents | The celebrated Lévy--Khintchine theorem is a fundamental limiting law that describes the growth rate of the denominators of the convergents in the continued fraction expansion of a Lebesgue-typical real number. In a recent breakthrough, Cheung and Chevallier \textit{(Annales scientifiques de l'ENS, 2024)} extended this theorem to higher dimensions.
In this paper, we resolve a conjecture of Y. Cheung and answer a question of Cheung and Chevallier concerning Lévy--Khintchine type theorems for arbitrary norms. We also establish a higher-dimensional analogue of the Doeblin--Lenstra law.
While our results are new in higher dimensions, they also yield significant improvements in the classical one-dimensional setting. Specifically, we revisit the Lévy--Khintchine theorem and the Doeblin--Lenstra law through the lens of Mahler's influential proposal to study Diophantine approximation on fractals. In particular, we prove these results for almost every point on the middle-third Cantor set. More broadly, our framework applies to a wide class of measures, including those supported on curves and on self-similar fractals generated by iterated function systems (IFS), and it also allows constraints on the selection of best approximates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15683 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Lévy-Khintchine Theorems and a Conjecture of Y. Cheung Aggarwal, Gaurav Ghosh, Anish Number Theory Dynamical Systems 11J13, 11J83, 37A17 The celebrated Lévy--Khintchine theorem is a fundamental limiting law that describes the growth rate of the denominators of the convergents in the continued fraction expansion of a Lebesgue-typical real number. In a recent breakthrough, Cheung and Chevallier \textit{(Annales scientifiques de l'ENS, 2024)} extended this theorem to higher dimensions. In this paper, we resolve a conjecture of Y. Cheung and answer a question of Cheung and Chevallier concerning Lévy--Khintchine type theorems for arbitrary norms. We also establish a higher-dimensional analogue of the Doeblin--Lenstra law. While our results are new in higher dimensions, they also yield significant improvements in the classical one-dimensional setting. Specifically, we revisit the Lévy--Khintchine theorem and the Doeblin--Lenstra law through the lens of Mahler's influential proposal to study Diophantine approximation on fractals. In particular, we prove these results for almost every point on the middle-third Cantor set. More broadly, our framework applies to a wide class of measures, including those supported on curves and on self-similar fractals generated by iterated function systems (IFS), and it also allows constraints on the selection of best approximates. |
| title | Generalized Lévy-Khintchine Theorems and a Conjecture of Y. Cheung |
| topic | Number Theory Dynamical Systems 11J13, 11J83, 37A17 |
| url | https://arxiv.org/abs/2408.15683 |