# Construction of Minimal Bracketing Covers for Rectangles

Michael Gnewuch

Arxiv ID: 0807.4446•Last updated: 9/21/2021

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 $\delta$.

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