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

TL;DR

This paper proves that the known upper bound on the discrepancy of s-dimensional Halton sequences is tight by establishing a positive lower limit, confirming the estimate's exactness.

Contribution

It establishes the lower bound of the discrepancy of Halton sequences, confirming the known upper bound is asymptotically optimal.

Findings

The discrepancy of Halton sequences grows at least as fast as a constant times (ln N)^s / N.

The upper bound on discrepancy is asymptotically tight.

The result confirms the optimality of the known discrepancy bounds.

Abstract

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: $$ \overline{\lim}_{ N \to \infty} N \ln^{-s}(N) D_N >0. $$