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

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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