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

In 1982, A. K. Lenstra, H. W. Lenstra, and L. Lovász 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.