Lévy-Khintchine Theorems: effective results and central limit theorems

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 ($\mathit{\text{Math.}}\mathit{\text{Annalen}}$, 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 ($\mathit{\text{Annales}}\mathit{\text{scientifiques}}\mathit{\text{de}}\mathit{\text{l’ENS}}$, 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.