# An Upper Bound of the Minimal Dispersion via Delta Covers

## TL;DR

This paper establishes an upper bound on the largest empty test set volume for point sets in high-dimensional cubes, using delta covers, with specific bounds for axis-parallel boxes and toroidal cases.

## Contribution

It introduces a new upper bound on minimal dispersion based on delta covers, applicable to various geometric test sets in high dimensions.

## Key findings

- Bound of (log |Γ_δ|)/n + δ for minimal dispersion
- Specific bounds for axis-parallel boxes: (4d/n) log(9n/d)
- Specific bounds for torus: (4d/n) log(2n)

## Abstract

For a point set of $n$ elements in the $d$-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers $n$, $d$ and under the assumption of a $delta$-cover with cardinality $\vert \Gamma_\delta \vert$ we prove that there is a point set, such that the largest volume of such a test set without any point is bounded by $\frac{\log \vert \Gamma_\delta \vert}{n} + \delta$. For axis-parallel boxes on the unit cube this leads to a volume of at most $\frac{4d}{n}\log(\frac{9n}{d})$ and on the torus to $\frac{4d}{n}\log (2n)$.