An upper bound on the minimal dispersion

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 $\varepsilon$.