Fuglede’s spectral set conjecture for convex polytopes.(English)Zbl 1377.42014

It is said that a convex polytope $$\Omega$$ in $$\mathbb R^d$$ is spectral if the space $$L^2(\Omega)$$ admits an orthogonal basis consisting of exponential functions. A conjecture, which goes back to Fuglede (1974), states that $$\Omega$$ is spectral if and only if it can tile the space by translations. It is known that if $$\Omega$$ tiles then it is spectral, but the converse was proved only in dimension $$d = 2$$, by Iosevich, Katz and Tao. Kolountzakis’s result asserts that if a convex polytope $$\Omega\subset\mathbb R^d$$ is spectral, then it must be centrally symmetric. It is proved in the paper under review that also all the facets of $$\Omega$$ are centrally symmetric. These conditions are necessary for $$\Omega$$ to tile by translations. The authors also develop an approach which allows them to prove that in dimension $$d = 3$$ any spectral convex polytope $$\Omega$$ indeed tiles by translations. This confirms Fuglede’s conjecture for convex polytopes in $$\mathbb R^3$$.

MSC:

 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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References:

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