Asymptotically optimal bracketing covers for anchored boxes with applications to star discrepancy

Abstract

Bracketing covers and $\delta$-covers provide finite discretizations of the anchored boxes that define the star discrepancy. Let $N_{[]}(d,\delta)$ and $N(d,\delta)$ denote the corresponding bracketing and covering numbers. We prove the lower bounds \[ N_{[]}(d,\delta)\ge \lceil \delta^{-d}\rceil, \qquad N(d,\delta)\ge \left\lceil \frac{d!}{d^d}\,\delta^{-d}\right\rceil. \] We give two explicit constructions of bracketing covers. For every fixed $d$, together with the lower bound they imply $N_{[]}(d,\delta)=(1+o_d(1))\delta^{-d}$ as $\delta\downarrow0$. A first construction uses box-dependent anisotropic local grids and gives simple explicit bounds. A second, homothetic logarithmic-shell construction again attains this coefficient and gives $\limsup_{d\to\infty}N_{[]}(d,\delta)^{1/d}\le\delta^{-1}+e+O(\delta)$ as $\delta\downarrow0$. Combining these finite estimates with Gnewuch's general bracketing bound and a Hoeffding--Bernstein chaining argument shows that, for every $d,n\in\mathbb N$, there exists an $n$-point set with star discrepancy at most $2.3463\sqrt{d/n}$. Consequently, $\lceil5.5052d\varepsilon^{-2}\rceil$ points suffice for star discrepancy at most $\varepsilon$.