A metrical lower bound on the star discrepancy of digital sequences

Abstract

In this paper we study uniform distribution properties of digital sequences over a finite field of prime order. In 1998 it was shown by Larcher that for almost all $s$-dimensional digital sequences the star discrepancy $D_N^{\ast }$ satisfies an upper bound of the form $D_N^{\ast }=O((\log N{)}^s(\log \log N{)}^{2+\epsilon })$ for any $\epsilon >0$. Generally speaking it is much more difficult to obtain good lower bounds for specific sequences than upper bounds. Here we show that Larchers result is best possible up to some $\log \log N$ term. More detailed, we prove that for almost all $s$-dimensional digital sequences the star discrepancy satisfies $D_N^{\ast }\ge c(q,s)(\log N{)}^s\log \log N$ for infinitely many $N \in \NN$, where $c(q,s)>0$ only depends on $q$ and $s$ but not on $N$.