On the lower bound of the discrepancy of Halton's sequence

Abstract

Let $(H_s(n){)}_{n\ge 1}$ be an $s-$dimensional Halton's sequence. Let $D_N$ be the discrepancy of the sequence $(H_s(n){)}_{n=1}^N$. It is known that $N{D}_N=O({\ln }^{s}N)$ as $N\to \infty$. In this paper we prove that this estimate is exact: $${\overline{\lim }}_{N\to \infty }N{\ln }^{-s}(N)D_N>0.$$