A note on the quality of simultaneous Diophantine approximations obtained by the LLL algorithm

TL;DR

This paper improves the understanding of how well the LLL algorithm performs in generating simultaneous Diophantine approximations, enhancing previous results in the field.

Contribution

It provides a significant improvement on existing bounds for the quality of approximations obtained by the LLL algorithm.

Findings

Enhanced bounds on approximation quality

Better theoretical guarantees for LLL-based approximations

Improved understanding of algorithmic performance in Diophantine approximation

Abstract

In 1982, A. K. Lenstra, H. W. Lenstra, and L. Lov\'asz introduced the first polynomial-time method to factor a nonzero polynomial $f \in \mathbb{Q}[x]$ into irreducible factors. This algorithm, now commonly referred to as the LLL Algorithm, can also be applied to compute simultaneous Diophantine approximations. We present a significant improvement of a result by Bosma and Smeets on the quality of simultaneous Diophantine approximations achieved by the LLL Algorithm.