Directional discrepancy in two dimensions

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.

Diophantine approximations and directional discrepancy of rotated lattices

“…This result was extended to all finite sets Ω by Cassels [8] in 1956 (with some generalizations by Davenport [10] In their previous work [6] the authors of the present paper had made an attempt to understand this question in the case of infinite sets Ω. Obviously, it is too optimistic to expect the same estimates in this situation, hence ψ(q) in (1.1) has to be an increasing function whose nature depends on the geometry of Ω. Generalizing the methods of Cassels and Davenport, we have considered several particular classes of sets: lacunary sequences, lacunary sets of finite order (see §3.3 for the definition), and sets with small upper Minkowski dimension.…”

“…For infinite rotation sets Ω one should anticipate the right-hand side to be somewhat smaller than 1/q 2 , which in turn would lead to larger discrepancy bounds depending on the geometry of Ω. Results of this type have been obtained in [6] for several particular examples of rotation sets.…”

“…It is therefore interesting to understand the exact nature of the relation between D Ω (N ) and the geometry of the set Ω. Several intermediate situations, such as lacunary sets (and their generalizations) and sets with small upper Minkowski dimension, have been considered in the prior work of the authors of this paper [6], where some upper discrepancy bounds for these partial cases have been obtained. The present paper generalizes previous results and provides a general method of obtaining upper bounds on directional discrepancy based on the geometric (entropy) properties of the set Ω.…”

“…Finally, using the approach described in [5,9,6], these estimates are applied to directional discrepancy in §5. Using this machinery one can obtain directional discrepancy estimates for any rotation set Ω; however the optimization required along the way prevents one from being able to write a generic formula for D Ω (N ) in terms of the covering function of Ω.…”

Applications of geometric discrepancy in numerical analysis and statistics

“…In this context, we mention one construction of explicit point sets in dimension 2 which considers more general test sets. Namely the construction by Bilyk, Ma, Pipher, Spencer [8] where discrepancy with respect to certain rotated boxes is considered.…”