Conjecture Is False in 5 and Higher Dimensions


We give an example of a set Ω ⊂ R which is a finite union of unit cubes, such that L2(Ω) admits an orthonormal basis of exponentials { 1 |Ω|1/2 ej ·x : ξj ∈ Λ} for some discrete set Λ ⊂ R , 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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@inproceedings{Tao2008ConjectureIF, title={Conjecture Is False in 5 and Higher Dimensions}, author={Terence Tao}, year={2008} }