论文标题
最终定期集作为最小添加剂补充
On Eventually Periodic Sets as Minimal Additive Complements
论文作者
论文摘要
我们说，如果还有其他一些子集合$ w $ $ g $的$ g $ g $ $ g $的子集$ c $，以最小的添加剂补充}，以至于$ c+w = \ {c+w = \ {c+w = in c，in c，\ w \ in w \} in w \} = g $ g $ and ye co $ c $ c $ c'+c'+c'在最近的论文中，伯克罗夫（Burcroff）和伦茨拉拉（Luntzlara）研究了，除其他方面，“最终定期集合”的条件是无限的无限工会（朝着积极的方向）算术进步和单胎，并且是在$ \ mathbb z $中的最小添加剂。在本文中，我们将进一步研究这个问题。我们以$ m $的界限形式给出了一些足够的条件，以最终定期设置为最小的添加剂补充；特别是我们表明“所有最终的周期集最终都是最小的添加剂补充”。此外，我们将其推广到一个框架，在该框架中，积分的“模式”降低到$ \ Mathbb Z $，我们证明，以这种方式出现的所有集合最终都是最小的添加剂补充。我们还介绍了形式主义的形式主义系列，该系列纯粹是簿记员写下证明。通过我们的工作，我们能够在大量案件中回答Burcroff和Luntzlara的问题。
We say a subset $C$ of an abelian group $G$ \textit{arises as a minimal additive complement} if there is some other subset $W$ of $G$ such that $C+W=\{c+w:c\in C,\ w\in W\}=G$ and such that there is no proper subset $C'\supset C$ such that $C'+W=G$. In their recent paper, Burcroff and Luntzlara studied, among many other things, the conditions under which "eventually periodic sets", which are finite unions of infinite (in the positive direction) arithmetic progressions and singletons, arise as minimal additive complements in $\mathbb Z$. In the present paper we shall study this question further. We give, in the form of bounds on the period $m$, some sufficient conditions for an eventually periodic set to be a minimal additive complement; in particular we show that "all eventually periodic sets are eventually minimal additive complements". Moreover, we generalize this to a framework in which "patterns" of points are projected down to $\mathbb Z$, and we show that all sets which arise this way are eventually minimal additive complements. We also introduce a formalism of formal power series, which serves purely as a bookkeeper in writing down proofs. Through our work we are able to answer a question of Burcroff and Luntzlara in a large class of cases.
