Asymptotically optimal bracketing covers for anchored boxes

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 also construct, for every fixed $d$, bracketing covers which, together with the lower bound, show that $N_{[]}(d,\delta)=(1+o_d(1))\delta^{-d}$ as $\delta\downarrow0$. The construction combines a coarse partition with box-dependent anisotropic local grids. Its shared vertices yield $\delta$-covers with asymptotic upper coefficient one. Explicit upper bounds are obtained for both quantities.