Fractal uncertainty for discrete 2D Cantor sets

We prove that a self-similar Cantor set in $\mathbb{Z}_N \times \mathbb{Z}_N$ has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of Lang's conjecture in number theory due to Ruppert and Beukers & Smyth. Our theorem answers a question of Dyatlov and has applications to open quantum maps.