Regularizing random points by deleting a few

 Suppose we want to distribute n points in the unit cube in such a way that any axis-parallel rectangle anchored in the origin (a natural generalization of the CDF) captures a number of points approximately proportional to its volume; how do we do it and what's the best we can hope for?  We'll discuss the state of the art and explain the idea behind the existing arguments. This leads to another natural problem: if someone gives us n iid uniformly distributed random points on the unit interval, how regular can we make the set by deleting (joint work with Bilyk '25) or moving (Smirnov-Vershynin '25) some of them?