Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

TL;DR

This paper investigates the properties of co-minimal pairs in groups, especially focusing on lacunary sequences in integers and higher-dimensional free abelian groups, revealing that many such sequences can form co-minimal pairs.

Contribution

It characterizes which subsets in abelian groups can be part of co-minimal pairs, especially showing that most lacunary sequences have this property and that uncountably many such subsets exist.

Findings

Most lacunary sequences can be part of co-minimal pairs.

Any infinite subset of a finitely generated abelian group has uncountably many co-minimal partners.

These sets can satisfy specific algebraic properties.

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.