Construction of Minimal Bracketing Covers for Rectangles

TL;DR

This paper constructs explicit minimal $oldsymbol{ ext{delta}}$-bracketing covers for rectangles in the 2D unit cube, achieving near-optimal cardinality bounds and analyzing their properties and explicit sizes.

Contribution

It provides explicit constructions of minimal $oldsymbol{ ext{delta}}$-bracketing covers for rectangles, matching theoretical lower bounds and analyzing their asymptotic behavior.

Findings

Cardinality of covers bounded by $oldsymbol{ ext{delta}}^{-2} + o( ext{delta}^{-2})$

Constructed covers are essentially optimal in 2D

Analyzed asymptotic expansion and explicit sizes of covers

Abstract

We construct explicit $\delta$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $\delta$-bracketing covers are bounded from above by $\delta^{-2} + o(\delta^{-2})$. A lower bound for the cardinality of arbitrary $\delta$-bracketing covers for $d$-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008) 154-172] implies the lower bound $\delta^{-2}+O(\delta^{-1})$ in dimension $d=2$, showing that our constructed covers are (essentially) optimal. We study also other $\delta$-bracketing covers for the set system of rectangles, deduce the coefficient of the most significant term $\delta^{-2}$ in the asymptotic expansion of their cardinality, and compute their cardinality for explicit values of $\delta$.