%0 Journal Article
%T Directional discrepancy in two dimensions
%A Dmitriy Bilyk
%A Xiaomin Ma
%A Jill Pipher
%A Craig Spencer
%J Mathematics
%D 2009
%I arXiv
%R 10.1112/blms/bdr050
%X In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. 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 finite 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.
%U http://arxiv.org/abs/0911.3971v1