Terence Tao gives an example of a set Ω ⊂ R^5, such that for L²(Ω), there exists an orthonaormal basis {|Ω|^(-1/2)e^(2πiξj∗x): ξj ∈ Λ} for some discrete set Λ ⊂ R^5, but Ω does not tile R^5 by translations.
This disproves one direction of Fuglede’s conjecture.
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