Minimal Dispersion of Large Volume Boxes in the Cube

Abstract

In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. The dispersion of a subset of the cube is the supremal volume over all axis parallel boxes in the cube which do not intersect the given subset. The minimal n-point dispersion is the infimal dispersion over all subsets of the cube containing n points. Define the large volume regime as the set of real volumes greater than 14 . In this note we work exclusively in the large volume setting. The construction presented in this paper yields a dimension independent upper bound which is an improvement on, and is proportional to the square root of the best known bound in this regime. We also show that some intermediate estimates are sharp, given that the dimension is taken to be larger than a specified volume-dependent constant.