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arXiv · 2504.00715

Lower bounds on the minimal dispersion of point sets via cover-free families

Matěj Trödler·Jan Volec·Jan Vybíral
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Abstract

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.

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Matěj Trödler, Jan Volec, Jan Vybíral. 2025-04-01. Lower bounds on the minimal dispersion of point sets via cover-free families. https://doi.org/10.1016/j.jco.2025.101974

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