Hiding in the crowd: asympstotic bounds on blocking sets

Abstract

We consider the problem of blocking all rays emanating from a unit disk U by a minimum number N_d of unit disks in the two-dimensional space, where each disk has at least a distance d to any other disk. We study the asymptotic behavior of N_d, as d tends to infinity. Using a regular ordering of disks on concentric circular rings we derive upper and lower bounds and prove that $\frac{\pi^2}{16} \leq \frac{N_d}{d^2} \leq \frac{\pi^2}{2}$, as d goes to infinity