The Pompeiu problem and discrete groups

Abstract

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.

On the discrete Fuglede and Pompeiu problems

“…In our paper we investigate the discrete version of the Pompeiu problem on finite abelian groups. We note that the discrete Pompeiu problem for infinite abelian groups was studied in [13,26,36].…”

The discrete Pompeiu problem on the plane

“…They have rational coordinates, so the distances between them are commensurable. Now f satisfies both(8) and(9) for every isometry σ, and thus, by Lemma 4.6, f ≡ 0.…”

“…Apparently, the first results concerning the discrete Pompeiu property appeared in [13], where the author considers the Pompeiu problem for finite subsets of Z [4], where it is proved that the set of vertices of a square has the discrete Pompeiu property with respect to the group of isometries. Recently, M. J. Puls [9] considered the discrete Pompeiu problem in groups.…”

“…Another generalization of the Putnam problem appeared in [4], where it is proved that the set of vertices of a square has the discrete Pompeiu property with respect to the group of isometries. Recently, M. J. Puls [9] considered the discrete Pompeiu problem in groups.…”

“…Proof. Note that if the torsion free rank of G is less than continuum, then this is a special case of [9,Theorem 3.1]. In the general case let H be the subgroup of G generated by E. Then H is a finitely generated torsion free Abelian group, and thus H is isomorphic to Z n for some finite n. By Zeilberger's theorem [13], E does not have the discrete Pompeiu property in H w.r.t.…”