General and Refined Montgomery Lemmata

Montgomery's Lemma on the torus $\mathbb{T}^d$ states that a sum of $N$ Dirac masses cannot be orthogonal to many low-frequency trigonometric functions in a quantified way. We provide an extension to general manifolds that also allows for positive weights: let $(M,g)$ be a smooth compact $d-$dimensional manifold without boundary, let $(\phi_k)_{k=0}^{\infty}$ denote the Laplacian eigenfunctions, let $\left\{ x_1, \dots, x_N\right\} \subset M$ be a set of points and $\left\{a_1, \dots, a_N\right\} \subset \mathbb{R}_{\geq 0}$ be a sequence of nonnegative weights. Then $$\sum_{k=0}^{X}{ \left| \sum_{n=1}^{N}{ a_n \phi_k(x_n)} \right|^2} \gtrsim_{(M,g)} \left(\sum_{i=1}^{N}{a_i^2} \right) \frac{ X}{(\log{X})^{\frac{d}{2}}}.$$ This result is sharp up to the logarithmic factor. Furthermore, we prove a refined spherical version of Montgomery's Lemma, and provide applications to estimates of discrepancy and discrete energies of $N$ points on the sphere $\mathbb{S}^{d}$.