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 ∈ (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] such that they intersect every axis-aligned box inside [0, 1] 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.