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Fuglede's conjecture is false in 5 and higher dimensions

Terence Tao
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Abstract

We give an example of a set $Ω\subset \R^5$ which is a finite union of unit cubes, such that $L^2(Ω)$ admits an orthonormal basis of exponentials $\{\frac{1}{|Ω|^{1/2}} e^{2πi ξ_j \cdot x}: ξ_j \in Λ\}$ for some discrete set $Λ\subset \R^5$, but which does not tile $\R^5$ by translations. This answers a conjecture of Fuglede in the negative, at least in 5 and higher dimensions.

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Terence Tao. 2003-06-09. Fuglede's conjecture is false in 5 and higher dimensions. https://arxiv.org/abs/math/0306134

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