General and Refined Montgomery Lemmata

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Montgomery's Lemma on the torus Tᵈ 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,, xₙ} ⊂ M be a set of points and {a₁,, aₙ} ⊂ R≥ ₀ be a sequence of nonnegative weights. Then ∑ₖ₌₀ˣ | ∑ₙ₌₁ⁿ aₙ ϕₖ(xₙ) |² ₍ₘ,g₎ (∑ᵢ₌₁ⁿaᵢ²) X(X)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 Sᵈ.