论文标题
作为最小添加剂补充在整数中产生的集合
Sets Arising as Minimal Additive Complements in the Integers
论文作者
论文摘要
Abelian Group $ G $的子集$ c $是$ w \ subseteq g $的最小添加剂补充，如果$ c + w = g $，如果$ c' + w \ w \ neq g $对于任何适当的子集$ c'\ subset c $。在本文中，我们研究了哪些整体集作为最小的添加剂补充。我们证实了权转的猜想，表明有限制的贝洛集有任意差距，以最小的添加剂补充出现。此外，我们的构造表明，任何此类集合都属于联合对，增强了Biswas和Saha的结果。我们绑定了在有限集合中以最小添加剂补充而产生的辛迪克集合的上和下部的Banach密度。我们提供了一些必要的条件，以最终定期出现作为最小的添加剂补充，并证明这些必要的条件也足以满足某些最终定期集合的类别。我们以有关最小添加剂补充的结构的几个猜想和问题结束。
A subset $C$ of an abelian group $G$ is a minimal additive complement to $W \subseteq G$ if $C + W = G$ and if $C' + W \neq G$ for any proper subset $C' \subset C$. In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that arise as minimal additive complements to finite sets. We provide some necessary conditions for an eventually periodic set to arise as a minimal additive complement and demonstrate that these necessary conditions are also sufficient for certain classes of eventually periodic sets. We conclude with several conjectures and questions concerning the structure of minimal additive complements.
