2025/12/17 by Suzuki, Kohei, Goda, Takashi
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.15007
We study the probabilistic existence of point configurations satisfying the (0, m, d)-net property in base b within a randomly generated point set in the d-dimensional unit cube. We first derive an upper bound on the number of geometric patterns for (0, m, d)-nets in base b. By applying the concentration inequalities together with this bound, we give lower and upper estimates for the probability that a set of N random points contains a (0, m, d)-net as a subset. This result leads to necessary and sufficient scaling conditions on N and m such that this probability converges to 1.