We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the discrete Pompeiu property. Using this description and the revealed connection we prove that Fuglede's conjecture holds for Z p n q 2 , where p and q are different primes. In particular, we show that every spectral subset of Z p n q 2 tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede's conjecture holds for Z 2 p .

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We investigate the functional equation n i=1 a i f (b i x + c i y) = 0, where a i , b i , c i ∈ C, and the unknown function f is defined on the field K = Q(b 1 , . . . , b n , c 1 , . . . , c n ). (It is easy to see that every solution on K can be extended to C as a solution.) Let S 1 denote the set of additive solutions defined on K. We prove that S 1 is spanned by S 1 ∩ D, where D is the set of the functions φ • D, where φ is a field automorphism of C and D is a differential operator on K. We say that the equation We show that S is spanned by S ∩ A, where A is the algebra generated by D. This implies that if S is translation invariant, then spectral synthesis holds in S. The main ingredient of the proof is the observation that if V is a variety on the Abelian group (K * ) k under multiplication, and every function F ∈ V is k-additive on K k , then spectral synthesis holds in V .We give several applications, and describe the set of solutions of equations having some special properties (e.g. having algebraic coefficients etc.).