Ioannis Parissis: The discrepancy function in two dimensions

Let A_N be an N-point set in the unit square and consider the discrepancy function

D_N(x) := #(A_N ∩ [0,x)) - N|[0,x)|

where x = (x₁,x₂) ∈ [0,1]², [0,x) = [0,x₁)×[0,x₂), and |[0,x)| denotes the Lebesgue measure of the rectangle. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points — Nx₁x₂ — in the rectangle. A basic theme of discrepancy theory is to study the “size” of this function in terms of N . It turns out that no matter how the N points are selected, their distribution must be far from uniform, i.e. the discrepancy function must be “large”. In this talk I will give an overview of some classical results in discrepancy theory that quantify the principle described above. I will also give an example of an extremal set for discrepancy, in particular the van der Corput point set. Finally, if time permits, I will discuss some size estimates for the discrepancy function obtained in a joint work with D. Bilyk, M. Lacey and A.Vagharshakyan. For example
we prove that

‖D_N‖_BMO ≳ (log N)^½.

This estimate is sharp. For the van der Corput set, we have ‖D_N‖_BMO ≲ (log N)^½, whenever N = 2ⁿ for some positive integer n.