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Generalized Lévy-Khintchine Theorems and a Conjecture of Y. Cheung

Generalized Lévy-Khintchine Theorems and a Conjecture of Y. Cheung

来源:Arxiv_logoArxiv
英文摘要

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.

Gaurav Aggarwal、Anish Ghosh

数学

Gaurav Aggarwal,Anish Ghosh.Generalized Lévy-Khintchine Theorems and a Conjecture of Y. Cheung[EB/OL].(2025-07-31)[2025-08-11].https://arxiv.org/abs/2408.15683.点此复制

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