We give an example of a set R 5 which is a finite union of unit cubes, such that L 2 () admits an orthonormal basis of exponentials { 1 || 1/2 e 2i j x : j } for some discrete set R 5 , but which does not tile R 5 by translations. This answers (one direction of) a conjecture of Fuglede [1] in the negative, at least in 5 and higher dimensions.
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Terence Tao (2004) studied this question.