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

We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real r is an element of (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 is an element of (0, 1) is fixed. (C) 2017 Elsevier Ltd. All rights reserved.