#### Vol. 13, No. 3, 2020

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On the discrete Fuglede and Pompeiu problems

### Gergely Kiss, Romanos Diogenes Malikiosis, Gábor Somlai and Máté Vizer

Vol. 13 (2020), No. 3, 765–788
##### Abstract

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 ${ℤ}_{{p}^{n}{q}^{2}}$, 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 ${ℤ}_{p}^{2}$.

##### Keywords
Pompeiu problem, spectral set conjecture, tiling, vanishing sums of roots of unity
##### Mathematical Subject Classification 2010
Primary: 39B32, 43A46
Secondary: 13F20, 20K01
##### Milestones
Received: 10 July 2018
Revised: 2 March 2019
Accepted: 8 April 2019
Published: 15 April 2020
##### Authors
 Gergely Kiss Faculty of Science Mathematics Research Unit University of Luxembourg Luxembourg Luxembourg Romanos Diogenes Malikiosis Department of Mathematics Aristotle University of Thessaloniki Thessaloniki Greece Gábor Somlai MTA-ELTE Geometric and Algebraic Combinatorics Research Group NKFIH 115799 and Faculty of Science, Institute of Mathematics Eötvös Loránd University Budapest Hungary Máté Vizer Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest Hungary