Asymptotically optimal bracketing covers for anchored boxes

Abstract

Bracketing covers and $δ$-covers provide finite discretizations of the anchored boxes that define the star discrepancy. Let $N_{[]}(d,δ)$ and $N(d,δ)$ denote the corresponding bracketing and covering numbers. We prove the lower bounds \[ N_{[]}(d,δ)\ge \lceil δ^{-d}\rceil, \qquad N(d,δ)\ge \left\lceil \frac{d!}{d^d}\,δ^{-d}\right\rceil. \] We also construct, for every fixed $d$, bracketing covers which, together with the lower bound, show that $N_{[]}(d,δ)=(1+o_d(1))δ^{-d}$ as $δ\downarrow0$. The construction combines a coarse partition with box-dependent anisotropic local grids. Its shared vertices yield $δ$-covers with asymptotic upper coefficient one. Explicit upper bounds are obtained for both quantities.

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“10 K. SUZUKI Declaration of generative AI use The author used ChatGPT 5.6 Sol for literature searches, exploratory development, and assistance in preparing portions of the exposition and LaTeX source. All mathematical arguments, calculations, references, and conclusions were ind”

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2026-08-23T19:17:18.697494Z

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