On the star discrepancy of sequences in the unit interval

TL;DR

This paper improves the lower bound estimate for the supremum constant related to the star discrepancy of sequences in the unit interval, advancing understanding of the distribution irregularities of sequences.

Contribution

It establishes a new lower bound for the supremum constant c* in the star discrepancy inequality, surpassing previous estimates.

Findings

New lower bound c* > 0.0646363 for star discrepancy constant.

Improved understanding of the distribution irregularities of sequences in [0,1).

Advances theoretical bounds in discrepancy theory.

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 $N D_N^* \ge c \cdot \log N$ holds for infinitely many $N$. Let $c^*$ be the supremum of all such $c$ with this property. We show $c^* > 0.0646363$, thereby improving the until now known estimates.