On the lower bound of the discrepancy of (t; s) sequences: I

Mordechay B. Levin

Published 2015-05-25Version 1

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