Improving a Constant in High-Dimensional Discrepancy Estimates

Abstract

For all $s\ge 1$ and $N\ge 1$ there exist sequences $(z_1,\dots ,z_N)$ in $[0,1{]}^s$ such that the star-discrepancy of these points can be bounded by $$D_N^{\ast }(z_1,\dots ,z_N)\le c\frac{\sqrt{s}}{\sqrt{N}}.$$ 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 $c=9$.