Consider an eigenfunction of the Laplacian on a torus. How small can its L²-norm be on small balls? We provide partial answers to this question by exploiting the distribution of integer points on spheres, basic properties of polynomials, and Nazarov--Tur\'an type estimates for exponential polynomials. Applications to quantum limits and control theory are given.
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Germain et al. (2024) studied this question.
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