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

TL;DR

This paper establishes a precise lower bound for the discrepancy of generalized Halton sequences, showing that the known upper bound is tight and cannot be improved.

Contribution

It proves that the discrepancy of generalized Halton sequences has a matching lower bound, confirming the optimality of the known upper bound.

Findings

Discrepancy of Halton sequences grows at least as fast as the known upper bound.

The lower bound matches the order of the upper bound, confirming its tightness.

Provides a constant C(H_s) depending on the sequence.

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 $$ \max_{1 \leq M \leq N} M \emph{D}^{*}_{M} \geq C(H_s) \log_2^s N \quad {\rm for} \; \; N=2,3,... \; . $$