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2017 The Fuglede conjecture holds in $\mathbb{Z}_p\times \mathbb{Z}_p$
Alexander Iosevich, Azita Mayeli, Jonathan Pakianathan
Anal. PDE 10(4): 757-764 (2017). DOI: 10.2140/apde.2017.10.757

Abstract

In this paper we study subsets E of pd such that any function f : E can be written as a linear combination of characters orthogonal with respect to E. We shall refer to such sets as spectral. In this context, we prove the Fuglede conjecture in p2, which says in this context that E p2 is spectral if and only if E tiles p2 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).

Citation

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Alexander Iosevich. Azita Mayeli. Jonathan Pakianathan. "The Fuglede conjecture holds in $\mathbb{Z}_p\times \mathbb{Z}_p$." Anal. PDE 10 (4) 757 - 764, 2017. https://doi.org/10.2140/apde.2017.10.757

Information

Received: 3 December 2015; Revised: 15 December 2015; Accepted: 11 March 2016; Published: 2017
First available in Project Euclid: 19 October 2017

zbMATH: 1361.05014
MathSciNet: MR3649367
Digital Object Identifier: 10.2140/apde.2017.10.757

Subjects:
Primary: 05A18, 11P99, 41A10, 42B05, 52C20

Rights: Copyright © 2017 Mathematical Sciences Publishers

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