Abstract
Let G be an additive and finite Abelian group, and p a prime number that does not divide the order of G. We show that if G has the universal spectrum property, then so does \(G\times {\mathbb {Z}}_p\). This is similar to and extends a previous result for cyclic groups using the same dilation trick but on non-cyclic groups as well. Inductively applying this statement on the known list of permissible G one can replace p with any square-free number that does not divide the order of G, and produce new tiling to spectral results in finite Abelian groups generated by at most two elements.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bhattacharya, S.: Periodicity and decidability of tilings of \(\mathbb{Z} ^2\). Am. J. Math. 142, 255–266 (2020)
Coven, E., Meyerowitz, A.: Tiling the integers with translates of one finite set. J. Algebra 212, 161–174 (1999)
Dutkay, D., Lai, C.: Some reduction of the spectral set conjecture on integers. Math. Proc. Camb. Philos. Soc. 156(1), 123–135 (2014)
Fallon, T., Kiss, G., Somlai, G.: Spectral sets and tiles in \({\mathbb{Z}}_p^2\times {\mathbb{ Z} }_q^2\). J. Funct. Anal. 282(12), 109472 (2022). https://doi.org/10.1016/j.jfa.2022.109472
Farkas, B., Matolcsi, M., Móra, P.: On Fuglede’s conjecture and the existence of universal spectra. J. Fourier Anal. Appl. 12(5), 483–494 (2006)
Fuglede, B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974)
Greenfeld, R., Tao, T.: The structure of translational tilings in \({\mathbb{ Z} }^d\). Discrete Anal. 16, 28 (2021)
Greenfeld, R., Tao, T.: A counterexample to the periodic tiling conjecture. Preprint. arXiv: 2211.15847 (2022)
Iosevich, A., Kolountzakis, M.: Periodicity of the spectrum in dimension one. Anal. PDE 6(4), 819–827 (2013)
Iosevich, A., Mayeli, A., Pakianathan, J.: The Fuglede conjecture holds in \({\mathbb{Z} }_p\times {\mathbb{Z} }_p\). Anal. PDE 10(4), 757–764 (2017)
Kenyon, R.: Rigidity of planar tilings. Invent. Math. 107, 637–651 (1992)
Kiss, G., Malikiosis, R., Somlai, G., Vizer, M.: Fuglede’s conjecture holds for cyclic groups of order \(pqrs\). J. Fourier Anal. Appl. 28, 79 (2022)
Kolountzakis, M., Lagarias, J.: Structure of tilings of the line by a function. Duke Math. J. 82, 653–678 (1996)
Kolountzakis, M., Matolcsi, M.: Complex Hadamard matrices and the spectral set conjecture. Collect. Math. 57, 281–291 (2006)
Kolountzakis, M., Matolcsi, M.: Tiles with no spectra. Forum Math. 18(3), 519–528 (2006)
Łaba, I.: The spectral set conjecture and multiplicative properties of roots of polynomials. J. Lond. Math. Soc. 65(3), 661–671 (2002)
Łaba, I., Londner, I.: Combinatorial and harmonic-analytic methods for integer tilings. Forum Math. Pi 10(8), 1–46 (2022)
Łaba, I., Londner, I.: Splitting for integer tilings and the Coven-Meyerowitz tiling conditions. Preprint arXiv: 2207.11809 (2022)
Łaba, I., Londner: The Coven-Meyerowitz tiling conditions for \(3\) odd prime factors. Invent. Math. (2022)
Lagarias, J., Wang, Y.: Tiling the line with translates of one tile. Invent. Math. 124, 341–365 (1996)
Lagarias, J., Wang, Y.: Spectral sets and factorizations of finite Abelian groups. J. Funct. Anal. 145, 73–98 (1997)
Lagarias, J., Szabó, S.: Universal spectra and Tijdeman’s conjecture on factorization of cyclic groups. J. Fourier Anal. Appl. 7, 63–70 (2001)
Lev, N., Matolcsi, M.: The Fuglede conjecture for convex domains is true in all dimensions. Acta Math. 228, 385–420 (2022)
Malikiosis, R.: On the structure of spectral and tiling subsets of cyclic groups. Forum Math. Sigma 10, 1–42 (2022)
Matolcsi, M.: Fuglede’s conjecture fails in dimension \(4\). Proc. Am. Math. Soc. 133(10), 3021–3026 (2005)
McMullen, P.: Convex bodies which tile space by translations. Mathematika 27, 113–121 (1980)
Sands, A.: Replacement of factors by subgroups in the factorization of Abelian groups. Bull. Lond. Math. Soc. 32(3), 297–304 (2000)
Shi, R.: Fuglede’s conjecture holds on cyclic groups \({\mathbb{Z}}_{pqr}\). Discrete Anal. 14 (2019). https://doi.org/10.19086/da.10570
Shi, R.: Equi-distribution on planes and spectral set conjecture on \({\mathbb{Z} }_{p^2}\times {\mathbb{Z} }_p\). J. Lond. Math. Soc. 102(2), 1030–1046 (2020)
Somlai, G.: Spectral sets in \({\mathbb{Z} }_{p^2qr}\) tile. Discrete Anal. 5, 10 (2023)
Szabó, S.: A type of factorization of finite Abelian groups. Discrete Math. 54, 121–124 (1985)
Szabó, S., Sands, A.: Factoring Groups into Subsets. CRC Press, Boca Raton (2008)
Tao, T.: Fuglede’s conjecture is false in \(5\) and higher dimensions. Math. Res. Lett. 11(2–3), 251–258 (2004)
Tijdeman, R.: Decomposition of the integers as a direct sum of two subsets. Number Theory Séminaire de Théorie des Nombres de Paris 1992–3, pp. 261–276. Cambridge University Press, Cambridge (1995)
Zhang, T.: Fuglede’s conjecture holds in \({\mathbb{Z}}_p\times {\mathbb{Z}}_{p^n}\). SIAM J. Discrete Math. 37(2), 1180–1197 (2023). https://doi.org/10.1137/22M1493598
Funding
Funded by the Jiangsu Education Department, General Program for Natural Science Research in Jiangsu Higher Institutions, with Grant No. 21KJD110001.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Stefan Steinerberger.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhou, W. Universal Spectra in \(G\times {\mathbb {Z}}_p\). J Fourier Anal Appl 30, 17 (2024). https://doi.org/10.1007/s00041-024-10074-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-024-10074-2