NTAG Seminar: A Generalized Levy-Khintchine Theorem

The classical Levy-Khintchine theorem describes the limiting distribution of the denominators of continued fraction convergents of a real number. In a recent breakthrough, Cheung and Chevallier extended this theorem to higher dimensions by considering best approximates of matrices. In this talk, I will present results that further generalize their work by introducing multiple natural notions of best approximates for matrices and proving Levy-Khintchine-type theorems in all these settings. Our results not only answer a question posed by Cheung and Chevallier about Levy-Khintchine-type theorems for arbitrary norms but also resolve a conjecture of Yitwah Cheung. Additionally, we extend the results of Cheung and Chevallier by proving their theorems for almost every point with respect to a broad class of measures, including fractal measures, while allowing best approximates to satisfy additional geometric and arithmetic constraints. This work extends recent results of Shapira and Weiss.