Metrical lower bounds on the discrepancy of digital Kronecker-sequences

Abstract

Digital Kronecker-sequences are a non-archimedean analog of classical Kronecker-sequences whose construction is based on Laurent series over a finite field. In this paper it is shown that for almost all digital Kronecker-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 the dimension $s$ and on the order $q$ of the underlying finite field, but not on $N$. This result shows that a corresponding metrical upper bound due to Larcher is up to some $\log \log N$ term best possible.