A note on minimal dispersion of point sets in the unit cube
share
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.
READ FULL TEXT