论文标题
最小补充和最大补充的反面问题
Inverse problems for minimal complements and maximal supplements
论文作者
论文摘要
给定一个Abelian Group $ G $的子集$ W $，如果$ W+C = G $，子集$ c $称为$ W $的添加剂补充；此外，如果没有适当的子集的$ c $具有此属性，那么我们说$ c $是$ w $的最小补充。自然要问哪些子集$ c $可以作为最小的$ W $进行的最小补充。我们表明，在有限的Abelian Group $ g $中，每个非空的子集$ c $ a size $ | c | \ \ leq 2^{2/3} | g |^{1/3}/（（（（（3e \ log | g |）^{2/3} $对于某些$ W $的最低补充。作为推论，我们推断出每个有限的非无标准子集的无限亚伯群互补的问题，也是一个最大的替补。补品。
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 $|C| \leq 2^{2/3}|G|^{1/3}/((3e \log |G|)^{2/3}$ 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.
