Close the gap between necessary and sufficient scaling conditions

Establish matching necessary and sufficient scaling conditions on the number of random points N and the net parameter m that guarantee, as m and N tend to infinity, convergence to 1 of the probability that an independently and uniformly sampled set of N points in the d-dimensional unit cube contains a (0,m,d)-net in base b as a subset.

Background

The paper studies when a random set of N points in the d-dimensional unit cube contains a structured (0,m,d)-net in base b. Under the assumptions that b is a prime power and d≤b+1, Theorem 3.1 provides a sufficient condition, N≥(1+ε)b^{md}m log b, and a separate necessary lower bound involving b^{md}(b!)^{m(d−1)/b}.

These bounds do not coincide in general, so the remaining problem is to determine the threshold or otherwise bridge the gap between the sufficient and necessary conditions for the probability of containing a (0,m,d)-net to converge to 1. The paper explicitly leaves this gap unresolved.