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Convex bodies with a point of curvature do not have Fourier bases

Alex Iosevich·Nets Hawk Katz·Terence Tao
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Abstract

We prove that no smooth symmetric convex body $Ω$ with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The non-symmetric case was proven by Kolountzakis). This is further evidence of Fuglede's conjecture, which states that such a basis is possible if and only if $Ω$ can tile $R^d$ by translations.

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Alex Iosevich, Nets Hawk Katz, Terence Tao. 1999-11-30. Convex bodies with a point of curvature do not have Fourier bases. https://arxiv.org/abs/math/9911167

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