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

Greenfeld, Rachel; Lev, Nir

Anal. PDE 10, No. 6, 1497-1538 (2017).

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 \(\mathbb R^3\).

Reviewer: Elijah Liflyand (Ramat-Gan)

MSC:

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

Keywords:

Fuglede’s conjecture; spectral set; tiling; convex polytope

References:

[1]	; Alexandrov, Trudy Mat. Inst. Steklov, 4, 87 (1933)
[2]	; Fedorov, Zap. Mineral. Obsc., 21, 1 (1885)
[3]	10.1016/0022-1236(74)90072-X · Zbl 0279.47014
[4]	10.1016/S0723-0869(01)80005-0 · Zbl 0990.42014
[5]	10.1016/j.jfa.2016.04.021 · Zbl 1348.42023
[6]	; Gruber, Convex and discrete geometry. Grundlehren der Mathematischen Wissenschaften, 336 (2007) · Zbl 1139.52001
[7]	10.4310/MRL.1999.v6.n2.a13 · Zbl 1007.42006
[8]	10.1353/ajm.2001.0003 · Zbl 0998.42001
[9]	10.4310/MRL.2003.v10.n5.a1 · Zbl 1087.42018
[10]	10.1007/BF01259371 · Zbl 1050.42016
[11]	; Kolountzakis, Illinois J. Math., 44, 542 (2000)
[12]	; Kolountzakis, Fourier analysis and convexity, 131 (2004)
[13]	; Kolountzakis, Gac. R. Soc. Mat. Esp., 13, 725 (2010)
[14]	; Kolountzakis, Illinois J. Math., 46, 1227 (2002)
[15]	10.1007/BF02771315 · Zbl 0203.54903
[16]	10.2307/1996000 · Zbl 0223.52007
[17]	10.1112/S0025579300010007 · Zbl 0432.52016
[18]	10.1112/S0025579300010238 · Zbl 0446.52006
[19]	10.1017/CBO9780511526282
[20]	10.4310/MRL.2004.v11.n2.a8 · Zbl 1092.42014
[21]	; Venkov, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Him., 9, 11 (1954)