Questions can be motivated from dynamical systems about the size of complements of a disjoint collection of Lagrangian tori in a symplectic manifold. We will discuss the simplest case, namely the complement of the integral product Lagrangians, $L(k,l)$ with $k,l \in \mathbb{N}$, inside $\mathbb{C}^2$. Here $L(k,l) = \{ |z_1| = k, |z_2|=l \}$. We will make some computations of the Gromov width and then describe joint work with Ely Kerman on the existence of Lagrangian tori in the complement.