Metrical lower bounds on the discrepancy of digital Kronecker-sequences

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 DN⁎⩾c(q,s)(logN)sloglogN for infinitely many N∈N, 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 loglogN term best possible.