On a sharp lemma of Cassels and Montgomery on manifolds

Abstract

Let $\left( \mathcal{M},g\right) $ be a $d$-dimensional compact connected Riemannian manifold and let $\left\{ \varphi_{m}\right\}_{m=0}^{+\infty}$ be a complete sequence of orthonormal eigenfunctions of the Laplace-Beltrami operator on $\mathcal{M}$. 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 $\mathcal{M}$, $\left\{ x\left( j\right) \right\}_{j=1}^{N}$, and positive weights $\left\{ a_{j}\right\}_{j=1}^{N}$ we have \[ \sum_{m=0}^{X} | \sum_{j=1}^{N} a_{j} \varphi_{m} ( x( j) ) | ^{2}\geq \max \{ CX\sum_{j=1}^{N}a_{j}^{2},( \sum_{j=1}^{N}a_{j}) ^{2}\}.\]