Improving a Constant in High-Dimensional Discrepancy Estimates

TL;DR

This paper improves the upper bound constant in high-dimensional discrepancy estimates from 10 to 9, enhancing the theoretical understanding of point distribution uniformity.

Contribution

The paper presents a tighter bound for the star-discrepancy constant in high dimensions, reducing it from 10 to 9, which advances discrepancy theory.

Findings

Bound on star-discrepancy constant improved to 9

Applicable for all dimensions s ≥ 1 and sample sizes N ≥ 1

Enhances theoretical understanding of uniform point distributions

Abstract

For all $s \geq 1$ and $N \geq 1$ there exist sequences $(z_1,\ldots,z_N)$ in $[0,1]^s$ such that the star-discrepancy of these points can be bounded by $$D_N^*(z_1,\ldots,z_N) \leq c \frac{\sqrt{s}}{\sqrt{N}}.$$ The best known value for the constant is $c=10$ as has been calculated by Aistleitner in \cite{Ais11}. In this paper we improve the bound to $c=9$.