Sharp discrepancy conjecture for arithmetic progressions in grids

Let $d\geq 1$ be an integer, and let $\mathbf{N}=(N_1,N_2,\cdots,N_d)$, where $N_1,\dots,N_d$ are positive integers. Write $\mathcal{A}_{\mathbf{N}}$ for the family of arithmetic progressions in the corresponding grid, and let $\disc(\mathcal{A}_{\mathbf{N}})$ denote its discrepancy. Sharp discrepancy conjecture.

$$\disc(\mathcal{A}_{\mathbf{N}})=\Theta_d\left(\max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^\frac{1}{2|I|+2}\right).$$

The conjecture asserts that the lower bound in Theorem~ is tight up to a constant depending only on $d$, removing the sub-logarithmic factor from the proved upper bound. The paper notes that the gap remains even for certain two-dimensional grids, including $\mathbf{N}=(N,\sqrt{N}(\log N)^k)$.