An improved bound for the star discrepancy of sequences in the unit interval

TL;DR

This paper establishes a new lower bound for the star discrepancy constant of sequences in the unit interval, slightly improving previous estimates and contributing to the understanding of uniform distribution measures.

Contribution

The paper improves the known lower bound for the star discrepancy constant, providing a tighter estimate for the minimal discrepancy growth rate of sequences.

Findings

New lower bound c* > 0.065664679... for star discrepancy constant.

Improved estimate refines understanding of sequence uniformity.

Results contribute to discrepancy theory and sequence analysis.

Abstract

It is known that there is a constant $c>0$ such that for every sequence $x_1, x_2,\ldots$ in $[0,1)$ we have for the star discrepancy $D^{*}_N$ of the first $N$ elements of the sequence that $N D^{*}_N\geq c\cdot \log N$ holds for infinitely many $N$. Let $c^{*}$ be the supremum of all such $c$ with this property. We show $c^{*}>0.065664679\ldots$, thereby slightly improving the estimates known until now.