# Levy-Khintchin Theorem for best simultaneous Diophantine approximations

## TL;DR

This paper extends key theorems from continued fraction theory to best simultaneous Diophantine approximations of vectors and matrices, establishing growth rates and distribution limits for approximation denominators.

## Contribution

It generalizes Levy-Khintchin and Bosma-Hendrik-Wiedijk theorems to higher dimensions and vectors, providing new almost sure results for Diophantine approximation sequences.

## Key findings

- Almost sure growth rate of denominators established.
- Distribution of product sequences characterized.
- Zero limit inferior for approximation sequences in higher dimensions.

## Abstract

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products $q_n d(q_n\theta, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real number $\theta$ by the ordinary continued fraction algorithm. Beside these two main results, we show that when $d\ge2$, for almost all vectors $\theta\in R^d$, $\liminf_{n\to\infty} q_{n+k}d(q_n\theta, Z^d)=0$ for all positive integers $k$, where $(q_n)_{n\in N}$ is the sequence of best approximation denominators of $\theta$.