Directional discrepancy in two dimensions
In this talk, we discuss some recent results on the geometric discrepancy with respect to families of rotated rectangles, which is joint work with D. Bilyk, X. Ma and C. Spencer. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of nite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy.