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arXiv · 1807.02844

On the discrete Fuglede and Pompeiu problems

Gergely Kiss·Romanos Diogenes Malikiosis·Gábor Somlai·Máté Vizer
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Abstract

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 $\mathbb{Z}_{p^n q^2}$, where $p$ and $q$ are different primes. In particular, we show that every spectral subset of $\mathbb{Z}_{p^n q^2}$ tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede's conjecture holds for $\mathbb{Z}_p^2$.

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Gergely Kiss, Romanos Diogenes Malikiosis, Gábor Somlai, Máté Vizer. 2019-04-11. On the discrete Fuglede and Pompeiu problems. https://doi.org/10.2140/apde.2020.13.765

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