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arXiv · 1507.08529

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

Mordechay B. Levin
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Abstract

Let $ (H_s(n))_{n \geq 1} $ be an $s-$dimensional generalized Halton's sequence. Let $\emph{D}^{*}_N$ be the discrepancy of the sequence $ (H_s(n) )_{n = 1}^{N} $. It is known that $D^{*}_{N} =O(\ln^s N)$ 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 $$ \max_{1 \leq M \leq N} M \emph{D}^{*}_{M} \geq C(H_s) \log_2^s N \quad {\rm for} \; \; N=2,3,... \; . $$

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Mordechay B. Levin. 2015-07-30. On the lower bound of the discrepancy of Halton's sequence II. https://arxiv.org/abs/1507.08529

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