Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions
Number Theory
2021-08-10 v1
Abstract
The study of minimal complements in a group or a semigroup was initiated by Nathanson. 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.
Keywords
Cite
@article{arxiv.2006.02429,
title = {Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions},
author = {Arindam Biswas and Jyoti Prakash Saha},
journal= {arXiv preprint arXiv:2006.02429},
year = {2021}
}