  Call a (strictly increasing) sequence $(r_{n})$ of natural numbers
\emph{regular} if it satisfies the following condition:
$r_{n+1}/r_{n}\to\theta\in\mathbb{R}^{>1}\cup\{\infty\}$ and, if $\theta$ is
algebraic, then $(r_{n})$ satisfies a linear recurrence relation whose
characteristic polynomial is the minimal polynomial of $\theta$. Our main
result states that $(\mathbb{Z},+,0,R)$ is superstable whenever $R$ is
enumerated by a regular sequence. We give two proofs of this result. One relies
on a result of E. Casanovas and M. Ziegler and the other on a quantifier
elimination result. We also show that $(\mathbb{Z},+,0,<,R)$ is NIP whenever
$R$ is enumerated by a regular sequence that is ultimately periodic modulo $m$
for all $m>1$.
