On the inverse of the star-discrepancy

Abstract

The inverse of the star-discrepancy $N^*(d,\ve)$ denotes the smallest possible cardinality of a set of points in $[0,1]^d$ achieving a star-discrepancy of at most $\ve$. By a result of Heinrich, Novak, Wasilkowski and Wo{ź}niakowski, $$ N^*(d,\ve) \leq c_{\textup{abs}} d \ve^{-2}. $$ Here the dependence on the dimension $d$ is optimal, while the precise dependence on $\ve$ is an open problem. In the present paper we prove that $$ N^*(d,\ve) \leq c_{\textup{abs}} d \ve^{-3/2} (\log (\ve^{-1}))^{1/2}. $$ This is a surprising result, which disproves a conjecture of Novak and Wo{ź}niakowski.