Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Spectra for cubes in products of finite cyclic groups

Authors: Elona Agora, Sigrid Grepstad and Mihail N. Kolountzakis
Journal: Proc. Amer. Math. Soc. 146 (2018), 2417-2423
MSC (2010): Primary 42B05, 52C22
Published electronically: February 21, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] Bálint Farkas and Szilárd Gy. Révész, Tiles with no spectra in dimension 4, Math. Scand. 98 (2006), no. 1, 44-52. MR 2221543
  • [3] Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101-121. MR 0470754
  • [4] 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
  • [5] Alex Iosevich, Azita Mayeli, and Jonathan Pakianathan, The Fuglede conjecture holds in $ \mathbb{Z}_p\times\mathbb{Z}_p$, Anal. PDE 10 (2017), no. 4, 757-764. MR 3649367
  • [6] Alex Iosevich and Steen Pedersen, Spectral and tiling properties of the unit cube, Internat. Math. Res. Notices 16 (1998), 819-828. MR 1643694
  • [7] Palle E. T. Jorgensen and Steen Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), no. 4, 285-302. MR 1700084
  • [8] Mihail N. Kolountzakis, Packing, tiling, orthogonality and completeness, Bull. London Math. Soc. 32 (2000), no. 5, 589-599. MR 1767712
  • [9] Mihail N. Kolountzakis and Máté Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collect. Math. Vol. Extra (2006), 281-291. MR 2264214
  • [10] Mihail N. Kolountzakis and Máté Matolcsi, Tiles with no spectra, Forum Math. 18 (2006), no. 3, 519-528. MR 2237932
  • [11] I. Łaba, Fuglede's conjecture for a union of two intervals, Proc. Amer. Math. Soc. 129 (2001), no. 10, 2965-2972. MR 1840101
  • [12] 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
  • [13] 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
  • [14] John Mackey, A cube tiling of dimension eight with no facesharing, Discrete Comput. Geom. 28 (2002), no. 2, 275-279. MR 1920144
  • [15] Máté Matolcsi, Fuglede's conjecture fails in dimension 4, Proc. Amer. Math. Soc. 133 (2005), no. 10, 3021-3026. MR 2159781
  • [16] Terence Tao, Fuglede's conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251-258. MR 2067470

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42B05, 52C22

Retrieve articles in all journals with MSC (2010): 42B05, 52C22

Additional Information

Elona Agora
Affiliation: Instituto Argentino de Matemática “Alberto P. Calderón” (IAM-CONICET), Argentina

Sigrid Grepstad
Affiliation: Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Austria
Address at time of publication: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Mihail N. Kolountzakis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, Voutes Campus, GR-700 13, Heraklion, Crete, Greece

Received by editor(s): February 9, 2016
Published electronically: February 21, 2018
Additional Notes: This work has been partially supported by (1) the “Aristeia II” action (Project FOURIERDIG) of the operational program Education and Lifelong Learning (co-funded by the European Social Fund and Greek national resources) and (for the last author only) (2) by grant No 4725 of the University of Crete.
The first author has been partially supported by Grants: MTM2013-40985-P, MTM2016-75196-P, PIP No. 112201501003553CO, UBACyT 20020130100422BA
The second author was currently supported by the Austrian Science Fund (FWF), Project F5505-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society