Jamison sequences in countably infinite discrete abelian groups

TL;DR

This paper extends the concept of Jamison sequences to countably infinite discrete abelian groups, providing an arithmetical characterization and identifying which sequences qualify as Jamison sequences.

Contribution

It generalizes Jamison sequences to a broader class of groups and characterizes them arithmetically, extending prior results from integers to general abelian groups.

Findings

The entire group sequence is a Jamison sequence.

Sequences generating subgroups of infinite index are not Jamison sequences.

Provides an arithmetical criterion for Jamison sequences in these groups.

Abstract

We extend the definition of Jamison sequences in the context of topological abelian groups. Then we study such sequences when the abelian group is discrete and countably infinite. An arithmetical characterization of such sequences is obtained, extending the result of Badea and Grivaux about Jamison sequences of integers. In particular, we prove that the sequence consisting of all elements of the group is a Jamison sequence. In the opposite, a sequence which generates a subgroup of infinite index in the group is never a Jamison sequence.