General and Refined Montgomery Lemmata

TL;DR

This paper extends Montgomery's Lemma from the torus to general manifolds, allowing for positive weights and providing sharp bounds with applications to discrepancy and energy estimates on spheres.

Contribution

It generalizes Montgomery's Lemma to arbitrary manifolds with positive weights and refines the spherical case, offering new tools for discrepancy and energy analysis.

Findings

Extended Montgomery's Lemma to general manifolds with weights

Proved a sharp lower bound involving eigenfunctions and eigenvalues

Applied results to discrepancy and energy estimates on spheres

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 $\left\{a_1, \dots, a_N\right\} \subset \mathbb{R}_{\geq 0}$ 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 $\mathbb{S}^{d}$.