Quasi-random points keep their distance

In contrast to random points that may cluster, quasi-random points keep their distance. These distances are investigated.1.If N independent random points in the n-dimensional unit hypercube are selected, two of these points may be arbitrarily close. However, if Q0, Q1, …, QN−1, are quasi-random points, the minimum distance between pairs of these points, dN, has a positive lower bound. For the Sobol sequence dN≥1/2nN−1. Numerical experiments suggest that for large Nequation(1)dN≍N−1/n.dN≍N−1/n.2.For certain search algorithms, it is important to know points Qi and Qi+1 that are not close. For the Sobol sequence, the distancesρ(Q2k,Q2k+1)=12n,and14n≤ρ(Q4k+1,Q4k+2)≤145n+c,where c = 0 for even n and c = 4 for odd n.3.Numerical estimations of dN for the Halton and Faure sequences were carried out. It is likely that for these sequences, (1) is true also.