Spectra for cubes in products of finite cyclic groups

Agora, Elona; Grepstad, Sigrid; Kolountzakis, Mihail

doi:10.1090/proc/14017

Spectra for cubes in products of finite cyclic groups
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by Elona Agora, Sigrid Grepstad and Mihail N. Kolountzakis PDF

Proc. Amer. Math. Soc. 146 (2018), 2417-2423 Request permission

Abstract:

We consider “cubes” in products of finite cyclic groups and we study their tiling and spectral properties. (A set in a finite group is called a tile if some of its translates form a partition of the group and is called spectral if it admits an orthogonal basis of characters for the functions supported on the set.) We show an analogue of a theorem due to Iosevich and Pedersen (1998), Lagarias, Reeds and Wang (2000), and the third author of this paper (2000), which identified the tiling complements of the unit cube in $\mathbb {R}^d$ with the spectra of the same cube.

References

Bálint Farkas, Máté Matolcsi, and Péter Móra, On Fuglede’s conjecture and the existence of universal spectra, J. Fourier Anal. Appl. 12 (2006), no. 5, 483–494. MR 2267631, DOI 10.1007/s00041-005-5069-7
Bálint Farkas and Szilárd Gy. Révész, Tiles with no spectra in dimension 4, Math. Scand. 98 (2006), no. 1, 44–52. MR 2221543, DOI 10.7146/math.scand.a-14982
Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754, DOI 10.1016/0022-1236(74)90072-x
Alex Iosevich, Nets Katz, and Terence Tao, The Fuglede spectral conjecture holds for convex planar domains, Math. Res. Lett. 10 (2003), no. 5-6, 559–569. MR 2024715, DOI 10.4310/MRL.2003.v10.n5.a1
Alex Iosevich, Azita Mayeli, and Jonathan Pakianathan, The Fuglede conjecture holds in $\Bbb {Z}_p\times \Bbb {Z}_p$, Anal. PDE 10 (2017), no. 4, 757–764. MR 3649367, DOI 10.2140/apde.2017.10.757
Alex Iosevich and Steen Pedersen, Spectral and tiling properties of the unit cube, Internat. Math. Res. Notices 16 (1998), 819–828. MR 1643694, DOI 10.1155/S1073792898000506
Palle E. T. Jorgensen and Steen Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), no. 4, 285–302. MR 1700084, DOI 10.1007/BF01259371
Mihail N. Kolountzakis, Packing, tiling, orthogonality and completeness, Bull. London Math. Soc. 32 (2000), no. 5, 589–599. MR 1767712, DOI 10.1112/S0024609300007281
Mihail N. Kolountzakis and Máté Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collect. Math. Vol. Extra (2006), 281–291. MR 2264214
Mihail N. Kolountzakis and Máté Matolcsi, Tiles with no spectra, Forum Math. 18 (2006), no. 3, 519–528. MR 2237932, DOI 10.1515/FORUM.2006.026
I. Łaba, Fuglede’s conjecture for a union of two intervals, Proc. Amer. Math. Soc. 129 (2001), no. 10, 2965–2972. MR 1840101, DOI 10.1090/S0002-9939-01-06035-X
Jeffrey C. Lagarias, James A. Reeds, and Yang Wang, Orthonormal bases of exponentials for the $n$-cube, Duke Math. J. 103 (2000), no. 1, 25–37. MR 1758237, DOI 10.1215/S0012-7094-00-10312-2
Jeffrey C. Lagarias and Peter W. Shor, Keller’s cube-tiling conjecture is false in high dimensions, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 279–283. MR 1155280, DOI 10.1090/S0273-0979-1992-00318-X
John Mackey, A cube tiling of dimension eight with no facesharing, Discrete Comput. Geom. 28 (2002), no. 2, 275–279. MR 1920144, DOI 10.1007/s00454-002-2801-9
Máté Matolcsi, Fuglede’s conjecture fails in dimension 4, Proc. Amer. Math. Soc. 133 (2005), no. 10, 3021–3026. MR 2159781, DOI 10.1090/S0002-9939-05-07874-3
Terence Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251–258. MR 2067470, DOI 10.4310/MRL.2004.v11.n2.a8