Infinite co-minimal pairs in the integers and integral lattices

Abstract

Given two nonempty subsets $A,B$ of a group $G$, they are said to form a co-minimal pair if $A\cdot B=G$, and $A^{\prime }\cdot B⊊G$ for any $\varnothing \ne A^{\prime }⊊A$ and $A\cdot B^{\prime }⊊G$ for any $\varnothing \ne B^{\prime }⊊B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\ge 1$, the group ${\mathbb{Z}}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $\sigma$, there exists a subset $A$ of ${\mathbb{Z}}^{2d}$ such that $A$ and $\sigma (A)$ form a co-minimal pair. The existence and construction of co-minimal pairs in the integers with both the subsets $A$ and $B$ ($A\ne B$) of infinite cardinality was unknown. We show that such pairs exist and explicitly construct these pairs satisfying a number of algebraic properties.

References in corpus (3)

• On minimal complements in groups
• On additive co-minimal pairs
• Asymptotic complements in the integers

Cited by in corpus (1)

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