On a sharp lemma of Cassels and Montgomery on manifolds
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Mathematische Annalen
On a sharp lemma of Cassels and Montgomery on manifolds Luca Brandolini1
· Bianca Gariboldi1 · Giacomo Gigante1
Received: 20 March 2020 / Revised: 26 October 2020 / Accepted: 3 November 2020 © The Author(s) 2020
Abstract Let (M, g) be a d-dimensional compact connected Riemannian manifold and let {ϕm }+∞ m=0 be a complete sequence of orthonormal eigenfunctions of the Laplace– Beltrami operator on M. We show that there exists a positive constant C such that for all integers N and X and for all finite sequences of N points in M, {x( j)} Nj=1 , and positive weights {a j } Nj=1 we have ⎧ 2 ⎛ ⎞2 ⎫ ⎪ ⎪ X N N ⎨ ⎬ N 2 ⎝ ⎠ . a j ϕm (x( j)) ≥ max C X aj, aj ⎪ ⎪ ⎩ ⎭ m=0 j=1 j=1 j=1 Mathematics Subject Classification 41A55 · 11K38 Let (M, g) be a d-dimensional compact connected Riemannian manifold, with normalized Riemannian measure μ such that μ(M) = 1, and Riemannian distance d(x, y). Let {λ2m }+∞ m=0 be the sequence of eigenvalues of the (positive) Laplace– Beltrami operator , listed in increasing order with repetitions, and let {ϕm }+∞ m=0 be an associated sequence of orthonormal eigenfunctions. In particular ϕ0 ≡ 1 and λ0 = 0.
Communicated by Loukas Grafakos. The authors have been supported by a GNAMPA 2019 project.
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Luca Brandolini [email protected] Bianca Gariboldi [email protected] Giacomo Gigante [email protected]
1
Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università degli Studi di Bergamo, Viale Marconi 5, Dalmine BG, Italy
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L. Brandolini et al.
This allows to define the Fourier coefficients of L 1 (M) functions as f (λm ) =
M
f (x)ϕm (x)dμ(x)
and the associated Fourier series +∞
f (λm )ϕm (x).
m=0
The main result of this paper is the following theorem. Theorem 1 There exists a positive constant C such that for all integers N and X and for all finite sequences of N points in M, {x( j)} Nj=1 , and positive weights {a j } Nj=1 we have ⎧ 2 ⎛ ⎞2 ⎫ ⎪ ⎪ X N N ⎨ ⎬ N 2 ⎝ ⎠ a ϕ (x( j)) ≥ max C X a , a . (1) j m j j ⎪ ⎪ ⎩ ⎭ m=0 j=1 j=1 j=1 Notice that the estimate 2 ⎛ ⎞2 X N N a j ϕm (x( j)) ≥ ⎝ aj⎠ m=0 j=1 j=1 is immediately obtained since for m = 0 one has ϕ0 (x) = 1 for all x in M. The essential part of the theorem is therefore the estimate 2 X N N ≥ CX a ϕ (x( j)) a 2j . j m m=0 j=1 j=1
(2)
2 Since for any m the expected value of a j ϕm (x( j)) is a 2j (see the proof of Proposition 2 below) the above estimate means that independently of how the points are chosen, there is a positive proportion of values of m between 0 and X for which a j ϕm (x( j))2 cannot be essentially smaller than its expected value. When M is the one-dimensional torus, the above theorem is classical and goes back to the work of Cassels [5]. He was interested in estimates on exponential sums, and their relation to Dirichlet’s approximation theorem. More precisely, as part of the proof of a slightly weaker vers
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