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

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 $\mathit{\text{(Annales}}\mathit{\text{scientifiques}}\mathit{\text{de}}\mathit{\text{l’ENS,}}\mathit{\text{2024)}}$ 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.