A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is an additive complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erd\H{o}s, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also study infinite sets forming co-minimal pairs. At the other extreme, motivated by unbounded arithmetic progressions in the integers, we look at sets which can never be a part of any co-minimal pair. This leads to a discussion on co-minimality, subgroups, approximate subgroups and asymptotic approximate subgroups of $G$.

中文翻译：

---

关于可加共极小对

如果$W+W'=G$，则阿贝尔群$G$ 中的一对非空子集$(W,W')$ 是可加补对。如果 $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$，则称 $W'$ 最小为 $W$。一般来说，给定一个群中的任意子集，最小补集的存在取决于它的结构。对偶问题询问给定这样的集合，它是否是某个子集的最小补充。自 Erd\H{o}s、Hanani、Lorentz 和其他人的时代以来，已经在整数表示的上下文中研究了加补。最小互补的概念是由 Nathanson 提出的。我们研究了互补对 $(W,W')$ 的紧密性，使得 $W$ 和 $W'$ 彼此最小。这些被称为共同极小对，我们证明任意自由阿贝尔群中的任何非空有限集都属于某个共同极小对。我们还研究了形成共同极小对的无限集。在另一个极端，受整数中无界算术级数的启发，我们研究永远不会成为任何共同极小对的一部分的集合。这导致了关于 $G$ 的共极小值、子群、近似子群和渐近近似子群的讨论。