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

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 \(\theta\).