Directional discrepancy in two dimensions

Details Directional discrepancy in two dimensions Directional discrepancy in two dimensions

2009-11-20

arxiv.org/abs/0911.3971v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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