General and refined Montgomery Lemmata

Montgomery’s Lemma on the torus 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 (ϕk)k=0∞ denote the Laplacian eigenfunctions, let { x₁, ⋯ , x_N} ⊂ M be a set of points and { a₁, ⋯ , a_N} ⊂ R_{≥ 0} be a sequence of nonnegative weights. Then, for all X≥ 0 , ∑k=0X|∑n=1Nanϕk(xn)|2≳(M,g)(∑i=1Nai2)X(logX)d2.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 S^d.