#### 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
##### Abstract

In this paper we study subsets $E$ of ${ℤ}_{p}^{d}$ such that any function $f:E\to ℂ$ 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 ${ℤ}_{p}^{2}$, which says in this context that $E\subset {ℤ}_{p}^{2}$ is spectral if and only if $E$ tiles ${ℤ}_{p}^{2}$ 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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