Stability and sparsity in sets of natural numbers

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.