Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori

Let Q(X) be any integral primitive positive definite quadratic form in k variables, where k≥ 4 , and discriminant D. For any integer n, we give an upper bound on the number of integral solutions of Q(X) = n in terms of n, k, and D. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus T^d for d≥ 5. This conjecture is motivated by the work of Berry [2,3] on the semiclassical eigenfunction hypothesis.