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Lower Bounds on the Minimal Dispersion of Point Sets via Cover-free Families by Trödler, Matěj; Volec, Jan; Vybíral, Jan is a scholarly article available to read on EtoBox.

What is Lower Bounds on the Minimal Dispersion of Point Sets via Cover-free Families about?

We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of $n$ points from $[0,1]^d$ and cover-free families from the extremal set theory. This connection was discovered in a recent paper of the authors. In this work, we apply a very recent result of Michel and Scott to obtain a whole range of new lower bounds on the number of points needed so that the largest volume of such a box is bounded by a given $\varepsilon$. Surprisingly, it turns out that for each of the new bounds, there is a choice of the parameters $d$ and $\varepsilon$ such that the bound outperforms the others.

Author

Trödler, Matěj; Volec, Jan; Vybíral, Jan

Published

2025

Language

EN