On the Dispersion of Sparse Grids

Abstract

For any natural number $d$ and positive number $\varepsilon$, we present a point set in the $d$-dimensional unit cube $[0,1]^d$ that intersects every axis-aligned box of volume greater than $\varepsilon$. These point sets are very easy to handle and in a vast range for $\varepsilon$ and $d$, we do not know any smaller set with this property.