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arXiv · 1505.04975

On the lower bound of the discrepancy of $(t,s)$ sequences: II

Mordechay B. Levin
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Abstract

Let $ (\bx(n))_{n \geq 1} $ be an $s-$dimensional Niederreiter-Xing sequence in base $b$. Let $D((\bx(n))_{n = 1}^{N})$ be the discrepancy of the sequence $ (\bx(n))_{n = 1}^{N} $. It is known that $N D((\bx(n))_{n = 1}^{N}) =O(\ln^s N)$ as $N \to \infty $. In this paper, we prove that this estimate is exact. Namely, there exists a constant $K>0$, such that $$ \inf_{\bw \in [0,1)^s} \sup_{1 \leq N \leq b^m} N D((\bx(n)\oplus \bw)_{n = 1}^{N}) \geq K m^s \quad {\rm for} \; \; m=1,2,...\;. $$ We also get similar results for other explicit constructions of $(t,s)$ sequences.

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Mordechay B. Levin. 2015-07-01. On the lower bound of the discrepancy of $(t,s)$ sequences: II. https://arxiv.org/abs/1505.04975

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