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

Abstract

Let $(H_s(n){)}_{n\ge 1}$ be an $s-$dimensional generalized Halton's sequence. Let ${\text{D}}_N^{\ast }$ be the discrepancy of the sequence $(H_s(n){)}_{n=1}^N$. It is known that $D_N^{\ast }=O({\ln }^{s}N)$ as $N\to \infty$. In this paper, we prove that this estimate is exact. Namely, there exists a constant $C(H_s)>0$, such that $$\underset{1\le M\le N}{\max }M{\text{D}}_M^{\ast }\ge C(H_s){\log }_2^{s}N\mathrm{for}N=2,3,....$$

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