Metrical lower bounds on the discrepancy of digital Kronecker-sequences

TL;DR

This paper establishes a lower bound on the star discrepancy of digital Kronecker-sequences, showing that their discrepancy behavior matches known upper bounds up to a logarithmic factor, for almost all sequences.

Contribution

It proves a metrical lower bound on the discrepancy of digital Kronecker-sequences, demonstrating the optimality of existing upper bounds up to a log log factor.

Findings

Star discrepancy is at least c(q,s) (log N)^s log log N for infinitely many N.

The lower bound applies to almost all digital Kronecker-sequences.

The result confirms the near-optimality of previously known upper bounds.

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.