On the vanishing of eigenfunctions of the Laplacian on tori

Consider an eigenfunction of the Laplacian on a torus. How small can its $L^2$-norm be on small balls? We provide partial answers to this question by exploiting the distribution of integer points on spheres, basic properties of polynomials, and Nazarov–Turán type estimates for exponential polynomials. Applications to quantum limits and control theory are given.