  Given a set $A\subseteq\mathbb{N}$, we consider the relationship between
stability of the structure $(\mathbb{Z},+,0,A)$ and sparsity of the set $A$. We
first show that a strong enough sparsity assumption on $A$ yields stability of
$(\mathbb{Z},+,0,A)$. Such sets include examples considered by Palac\'{i}n and
Sklinos and Poizat, many classical linear recurrence sequences (e.g. the
Fibonaccci numbers), and any set in which the limit of ratios of consecutive
elements diverges. Finally, we consider sparsity conclusions on sets
$A\subseteq\mathbb{N}$, which follow from model theoretic assumptions on
$(\mathbb{Z},+,0,A)$. We use a result of Erd\H{o}s, Nathanson, and
S\'{a}rk\"{o}zy to show that if $(\mathbb{Z},+,0,A)$ does not define the
ordering on $\mathbb{Z}$, then the lower asymptotic density of any finitary
sumset of $A$ is zero. Finally, in a theorem communicated to us by Goldbring,
we use a result of Jin to show that if $(\mathbb{Z},+,0,A)$ is stable, then the
upper Banach density of any finitary sumset of $A$ is zero.
