On the inverse of the star-discrepancy

TL;DR

This paper establishes a new upper bound on the inverse star-discrepancy, showing it grows proportionally to d times /2 power of v and a square root of the logarithm, challenging previous conjectures.

Contribution

The paper proves a novel upper bound on the inverse star-discrepancy, improving understanding of its dependence on v and disproving a prior conjecture.

Findings

New upper bound: N^*(d,v)  /2 (\u001og(v^{-1}))^{1/2}

Disproves previous conjecture by Novak and Wozniakowski

Enhances understanding of discrepancy behavior in high dimensions

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{\'z}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{\'z}niakowski.