Title:
Directional discrepancy in two dimensions
Abstract:
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 wellknown extremal cases are the axisparallel 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.
