Directional discrepancy in two dimensions
Classical Analysis and ODEs
2014-02-26 v1 Number Theory
Abstract
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.
Keywords
Cite
@article{arxiv.0911.3971,
title = {Directional discrepancy in two dimensions},
author = {Dmitriy Bilyk and Xiaomin Ma and Jill Pipher and Craig Spencer},
journal= {arXiv preprint arXiv:0911.3971},
year = {2014}
}