  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 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.
