Lower bounds for the directional discrepancy with respect to an interval of rotations

TL;DR

This paper establishes a lower bound on the directional discrepancy for rectangles rotated within a limited interval of directions, demonstrating that the discrepancy grows at least as fast as a power of the number of points.

Contribution

It provides a new lower bound for the discrepancy of rectangles with restricted rotation angles, extending understanding of geometric discrepancy in constrained settings.

Findings

Lower bound for discrepancy is at least N^{1/5} for restricted rotations

Constant in the bound depends on the interval of directions

Results apply to rectangles rotated within an interval less than π/4

Abstract

We show that the lower bound for the optimal directional discrepancy with respect to the class of rectangles in $\mathbb{R}^2$ rotated in a restricted interval of directions $[-\theta, \theta]$ with $\theta < \frac{\pi}{4}$ is of the order at least $N^{1/5}$ with a constant depending on $\theta$.