On the discrete Fuglede and Pompeiu problems

We investigate the discrete Fuglede conjecture and the 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 [math] , where [math] and [math] are different primes. In particular, we show that every spectral subset of [math] tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede’s conjecture holds for [math] .