Directional discrepancy in two dimensions

TL;DR

This paper investigates the geometric discrepancy of rotated rectangles in two dimensions, exploring intermediate cases between axis-aligned and fully rotated rectangles, and introduces methods to construct rotations with small discrepancy.

Contribution

It extends Davenport's lemma to construct lattice rotations with small discrepancy for various sets of directions, including lacunary sequences and sets with small Minkowski dimension.

Findings

Small discrepancy rotations for lacunary sequences of directions

Extensions of Davenport's lemma for different directional sets

Construction methods for lattice rotations with minimal discrepancy

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.