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 ${\left\{{\phi }_m\right\}}_{m=0}^{+\infty }$ 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 \[ \sum_{m=0}^{X} | \sum_{j=1}^{N} a_{j} φ_{m} ( x( j) ) | ^{2}\geq \max \{ CX\sum_{j=1}^{N}a_{j}^{2},( \sum_{j=1}^{N}a_{j}) ^{2}\}.\]