Abstract
In 2011, Nathanson proposed several questions on minimal complements in a group or a semigroup. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.
| Originalsprog | Engelsk |
|---|---|
| Tidsskrift | Ramanujan Journal |
| Vol/bind | 57 |
| Sider (fra-til) | 1445–1462 |
| ISSN | 1382-4090 |
| DOI | |
| Status | E-pub ahead of print - 2022 |
| Udgivet eksternt | Ja |
Bibliografisk note
Funding Information:The work of the first author was supported by the ISF Grant No. 662/15. The second author was supported by the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPRE Faculty Award IFA18-MA123 from the Department of Science and Technology, Government of India.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.