A class of non-convex polytopes that admit no orthonormal basis of exponentials

Abstract

A conjecture of Fuglede states that a bounded measurable set Ω ⊂ R, of measure 1, can tile R by translations if and only if the Hilbert space L(Ω) has an orthonormal basis consisting of exponentials eλ(x) = exp 2πi〈λ, x〉. If Ω has the latter property it is called spectral. Let Ω be a polytope in R with the following property: there is a direction ξ ∈ S such… (More)

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