Asymptotic complements in the integers

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 $C\setminus \{c\}$ is not an asymptotic complement $\mathrm{\forall }c\in C$. Asymptotic complements have been studied in the context of representations of integers since the time of Erd\H{o}s, 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.