An upper bound on the minimal dispersion

TL;DR

This paper establishes an upper bound on the minimal dispersion of point sets in high-dimensional cubes, showing that for sufficiently large N, there exists a set intersecting all axis-aligned boxes of volume ε, thus controlling the dispersion.

Contribution

The paper proves an explicit upper bound on the minimal dispersion in high dimensions, improving understanding of point distributions with small dispersion.

Findings

Existence of point sets with dispersion at most ε for large N

Explicit bound relating N, d, and ε

Improved theoretical understanding of dispersion in high dimensions

Abstract

For $\varepsilon\in(0,1/2)$ and a natural number $d\ge 2$, let $N$ be a natural number with \[ N \,\ge\, 2^9\,\log_2(d)\, \left(\frac{\log_2(1/\varepsilon)}{\varepsilon}\right)^2. \] We prove that there is a set of $N$ points in the unit cube $[0,1]^d$, which intersects all axis-parallel boxes with volume $\varepsilon$. That is, the dispersion of this point set is bounded from above by $\varepsilon$.