Abstract
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, 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 Sd.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1283-1297 |
| Number of pages | 15 |
| Journal | Mathematische Annalen |
| Volume | 373 |
| Issue number | 3-4 |
| DOIs | |
| State | Published - Apr 1 2019 |
Bibliographical note
Funding Information:Acknowledgements Parts of this work were started at the Workshop “Discrepancy Theory and Quasi-Monte Carlo methods” held at the Erwin Schrödinger Institute, September 25–29, 2017. The authors gratefully acknowledge its hospitality. Bilyk’s work is supported by NSF Grant DMS 1665007. Dai was supported by NSERC Canada under the Grant RGPIN 04702 Dai.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.