A note on minimal dispersion of point sets in the unit cube

Abstract

We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real r∈(0,1) and an integer d≥2, let N(r,d) denote the minimum number of points inside the d-dimensional unit cube [0,1]d such that they intersect every axis-aligned box inside [0,1]d of volume greater than r. We prove an upper bound on N(r,d), matching a lower bound of Aistleitner et al. up to a multiplicative constant depending only on r. This fully determines the rate of growth of N(r,d) if r∈(0,1) is fixed.