A metrical lower bound on the star discrepancy of digital sequences

TL;DR

This paper establishes a nearly tight lower bound on the star discrepancy of digital sequences, showing that for almost all such sequences, the discrepancy cannot be smaller than a certain order involving logarithmic factors.

Contribution

It proves that Larcher's upper bound on star discrepancy is essentially optimal by providing a matching lower bound for almost all digital sequences.

Findings

Star discrepancy for almost all digital sequences is at least c(q,s) (log N)^s log log N.

Larcher's upper bound is tight up to a log log N factor.

The lower bound applies infinitely often for N.

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 $D_N^\ast=O((\log N)^s (\log \log N)^{2+\varepsilon})$ for any $\varepsilon>0$. 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 $N$.