The two-sided Pompeiu problem for discrete groups

TL;DR

This paper investigates a two-sided Pompeiu problem in discrete groups, establishing conditions under which finite sets and entire groups possess the Pompeiu property using advanced algebraic techniques.

Contribution

It provides necessary and sufficient conditions for finite sets and groups to have the Pompeiu property in the context of discrete groups, employing group von Neumann algebra methods.

Findings

Characterization of finite sets with the Pompeiu property

Conditions for groups to be Pompeiu groups

Application of von Neumann algebra techniques

Abstract

We consider a two-sided Pompeiu type problem for a discrete group $G$. We give necessary and sufficient conditions for a finite set $K$ of $G$ to have the $\mathcal{F}(G)$-Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for $G$ to be a $\ell^2(G)$-Pompeiu group