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 $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B' \subsetneq B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\geq 1$, 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 $B$ ($A\neq B$) of infinite cardinality was unknown. We show that such pairs exist and explicitly construct these pairs satisfying a number of algebraic properties.