Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to ask which subsets $C$ can arise as minimal complements for some $W$. We show that in a finite abelian group $G$, every non-empty subset $C$ of size $|C| \leq |G|^{1/3}/(\log_2 |G|)^{2/3}$ is a minimal complement for some $W$. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for "dual" problems about maximal supplements.

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最小补和最大补的逆问题

给定阿贝尔群$G$的子集$W$，如果$W+C=G$，则子集$C$被称为$W$的加补；此外，如果 $C$ 的真子集没有这个性质，那么我们说 $C$ 是 $W$ 的最小补充。很自然地会问哪些子集 $C$ 可以作为某些 $W$ 的最小补充出现。我们证明在有限阿贝尔群 $G$ 中，每个大小为 $|C| 的非空子集 $C$ \leq |G|^{1/3}/(\log_2 |G|)^{2/3}$ 是一些 $W$ 的最小补充。作为推论，我们推导出无限阿贝尔群的每个有限非空子集都是最小补集。我们还为关于最大补充剂的“双重”问题得出了几个类似的结果。