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

Let $(H_s(n))_{n \geq 1}$ 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 $ND_N =O(\ln^s N)$ as $N \to \infty$. In this paper we prove that this estimate is exact: $$\overline{\lim}_{ N \to \infty} N \ln^{-s}(N) D_N >0.$$