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

Dmitriy Bilyk, Michelle Mastrianni

Published 2022-09-24Version 1

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 $\theta$.