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.
Bibliographical noteFunding Information:
of Mathematics for the warm welcome during the workshop ‘Small ball inequalities in analysis, probability, and irregularities of distribution’. Dmitriy Bilyk and Craig Spencer would like to thank the Institute for Advanced Study for its hospitality. Craig Spencer was also supported by the NSA Young Investigators Grant. In addition, the authors are indebted to William Chen and Giancarlo Travaglini for numerous interesting and fruitful discussions. The first and the third authors also express gratitude to Centre de Recerca Matemàtica in Barcelona for hospitality and support.
The authors acknowledge the support of the National Science Foundation (the first author, NSF grant DMS-0801036, the second and the third authors, NSF grant DMS-0901139). In addition, the first and the fourth authors were supported by the NSF grant 0635607 at the Institute for Advanced Study.