Generalized L\'{e}vy-Khintchine Theorems and a Conjecture of Y. Cheung

TL;DR

This paper extends the Lévy–Khintchine theorem to higher dimensions, resolves a conjecture of Cheung, and applies these results to fractals like the Cantor set, improving understanding of Diophantine approximation in complex settings.

Contribution

It proves a conjecture of Cheung regarding Lévy–Khintchine theorems for arbitrary norms and establishes a higher-dimensional analogue of the Doeblin–Lenstra law.

Findings

Resolved a conjecture of Y. Cheung on Lévy–Khintchine theorems.

Extended the Doeblin–Lenstra law to higher dimensions.

Proved classical results for almost every point on the middle-third Cantor set.

Abstract

The celebrated L\'evy--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 L\'evy--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 L\'evy--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.