On a sharp lemma of Cassels and Montgomery on manifolds

Abstract

AbstractLet $$( \mathcal {M},g ) $$ ( M , g ) be a d-dimensional compact connected Riemannian manifold and let $$\{ \varphi _{m}\} _{m=0}^{+\infty }$$ { φ m } m = 0 + ∞ be a complete sequence of orthonormal eigenfunctions of the Laplace–Beltrami operator on $$\mathcal {M}$$ 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 $$\mathcal {M}$$ M , $$\{x( j)\} _{j=1}^{N}$$ { x ( j ) } j = 1 N , and positive weights $$\{ a_{j}\} _{j=1}^{N}$$ { a j } j = 1 N we have $$\begin{aligned} \sum _{m=0}^{X}\left| \sum _{j=1}^{N}a_{j}\varphi _{m}( x( j)) \right| ^{2}\ge \max \left\{ CX\sum _{j=1}^{N}a_{j} ^{2},\left( \sum _{j=1}^{N}a_{j}\right) ^{2}\right\} . \end{aligned}$$ ∑ m = 0 X ∑ j = 1 N a j φ m ( x ( j ) ) 2 ≥ max C X ∑ j = 1 N a j 2 , ∑ j = 1 N a j 2 .