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. Let d>1. The dispersion of T⊂[0,1]d is defined as the supremum of the volume taken over all axis parallel boxes in the cube which do not intersect T. The minimal dispersion of n points in the cube is defined as the infimum of the dispersion taken over all T such that |T|=n. Define the “large volume” regime as the class of all volumes 14<r≤12. The inverse of the minimal dispersion is denoted as N(r,d). When the volume is large, the best known upper bound on N(r,d) is of the order (r−14)−1. The construction presented in this note yields an upper bound given byN(r,d)≤⌊πr−14⌋−3. Some of our intermediate estimates are sharp given the condition that d≥Cr, where Cr is a positive constant which depends only on the volume r.