Fuglede's conjecture is false in 5 and higher dimensions
Tao, Terence
Original · EN
We give an example of a set Ω⊂ ⁵ which is a finite union of unit cubes, such that L²(Ω) admits an orthonormal basis of exponentials {1|Ω|¹/² e²πⁱ ξʲ · ˣ: ξⱼ ∈ Λ} for some discrete set Λ⊂ ⁵, but which does not tile ⁵ by translations. This answers a conjecture of Fuglede in the negative, at least in 5 and higher dimensions.
English translation
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