On a sharp lemma of Cassels and Montgomery on manifolds

Let (M, g) be a d-dimensional compact connected Riemannian manifold and let {φm}m=0+∞ be a complete sequence of orthonormal eigenfunctions of the Laplace–Beltrami operator on M. We show that there exists a positive constant C such that for all integers N and X and for all finite sequences of N points in M, {x(j)}j=1N, and positive weights {aj}j=1N we have ∑m=0X|∑j=1Najφm(x(j))|2≥max{CX∑j=1Naj2,(∑j=1Naj)2}.