General and refined Montgomery Lemmata

Abstract

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 $\left\{ a_1, \dots , a_N\right\} \subset \mathbb {R}_{\ge 0}$ be a sequence of nonnegative weights. Then, for all $X \ge 0$,

$$\begin{aligned} \sum _{k=0}^{X}{ \left| \sum _{n=1}^{N}{ a_n \phi _k(x_n)} \right| ^2} \gtrsim _{(M,g)} \left( \sum _{i=1}^{N}{a_i^2} \right) \frac{ X}{(\log {X})^{\frac{d}{2}}}. \end{aligned}$$

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 $\mathbb {S}^{d}$.