The Pompeiu Problem and Discrete Groups

TL;DR

This paper explores the Pompeiu problem within discrete groups, providing conditions for the property in abelian and nonabelian free groups, and linking harmonicity to mean-value properties over spheres.

Contribution

It introduces a discrete group formulation of the Pompeiu problem, establishing necessary and sufficient conditions for abelian groups and extending results to nonabelian free groups.

Findings

Conditions for the Pompeiu property in abelian groups with low torsion free rank

Extension of Pompeiu problem results to nonabelian free groups

A sufficient condition for harmonicity based on mean-value over spheres

Abstract

We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.