On the discrete Fuglede and Pompeiu problems

TL;DR

This paper explores the discrete Fuglede and Pompeiu problems on finite abelian groups, establishing a connection and proving Fuglede's conjecture for specific groups, with implications for spectral sets and tiling.

Contribution

It introduces a geometric condition for the Pompeiu property, links it to Fuglede's conjecture, and proves the conjecture for groups of the form Z_{p^n q^2} and Z_p^2.

Findings

Fuglede's conjecture holds for Z_{p^n q^2} where p,q are primes.

Every spectral subset of Z_{p^n q^2} tiles the group.

Fuglede's conjecture holds for Z_p^2.

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 $\mathbb{Z}_p^2$.