Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be a minimal asymptotic complement if $C$ is an asymptotic complement, but $C\setminus \lbrace c\rbrace$ is not an asymptotic complement $\forall c\in C$. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in $\mathbb{Z}$ and deal with a problem of Nathanson on their existence and their inexistence.

中文翻译：

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整数中的渐近补

令 $W\subseteq \mathbb{Z}$ 是整数的非空子集。如果 $W+C$ 包含除有限大小的集合之外的几乎所有整数，则称非空集 $C\subseteq \mathbb{Z}$ 是 $W$ 的渐近补集。如果 $C$ 是渐近补，则称 $C$ 是最小渐近补，但 $C\setminus \lbrace c\rbrace$ 不是渐近补 $\forall c\in C$。自 Erdős、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渐近补，而最小渐近补的概念是由 Nathanson 提出的。在本文中，我们研究了 $\mathbb{Z}$ 中的最小渐近补，并处理了 Nathanson 关于它们存在和不存在的问题。