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arXiv · 0911.3971

Directional discrepancy in two dimensions

Dmitriy Bilyk·Xiaomin Ma·Jill Pipher·Craig Spencer
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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.

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Dmitriy Bilyk, Xiaomin Ma, Jill Pipher, Craig Spencer. 2009-11-20. Directional discrepancy in two dimensions. https://doi.org/10.1112/blms%2Fbdr050

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