Vol. 10, No. 4, 2017

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The Fuglede conjecture holds in $\mathbb{Z}_p\times \mathbb{Z}_p$

Alex Iosevich, Azita Mayeli and Jonathan Pakianathan

Vol. 10 (2017), No. 4, 757–764

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).

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exponential bases, Erdős problems, Fuglede conjecture
Mathematical Subject Classification 2010
Primary: 05A18, 11P99, 41A10, 42B05, 52C20
Received: 3 December 2015
Revised: 15 December 2015
Accepted: 11 March 2016
Published: 9 May 2017
Alex Iosevich
Department of Mathematics
University of Rochester
Rochester, NY 14627
United States
Azita Mayeli
Department of Mathematics and Computer Science
Queensborough Community College
222-05 56th Ave.
Bayside, NY 11364
United States
Jonathan Pakianathan
Department of Mathematics
University of Rochester
Rochester, NY 14627
United States