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The Fuglede
conjecture holds in $\mathbb{Z}_p\times \mathbb{Z}_p$
Alex Iosevich, Azita Mayeli and Jonathan Pakianathan |
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Vol. 10 (2017), No. 4, 757–764
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Abstract |
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In this paper we study subsets of such that any function can be written as a linear combination of characters orthogonal with respect to . We shall refer to such sets as spectral. In this context, we prove the Fuglede conjecture in , which says in this context that is spectral if and only if tiles by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof. The proof relies to a significant extent on the analysis of direction sets of Iosevich et al. (Integers 11 (2011), art. id. A39) and the tiling results of Haessig et al. (2011). |
Keywords
exponential bases, Erdős problems, Fuglede conjecture
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Mathematical Subject Classification 2010
Primary: 05A18, 11P99, 41A10, 42B05, 52C20
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Milestones
Received: 3 December 2015
Revised: 15 December 2015
Accepted: 11 March 2016
Published: 9 May 2017
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Authors |
| Alex Iosevich | |
| Department of Mathematics University of Rochester Rochester, NY 14627 United States |
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| Azita Mayeli | |
| Department of Mathematics and
Computer Science Queensborough Community College 222-05 56th Ave. Bayside, NY 11364 United States |
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| Jonathan Pakianathan | |
| Department of Mathematics University of Rochester Rochester, NY 14627 United States |
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