L\'{e}vy-Khintchine Theorems: effective results and central limit theorems

Abstract

The L\'evy-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 (\textit{Math. 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 L\'evy-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 (\textit{Annales scientifiques de 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.