Call a (strictly increasing) sequence $(r_n)$ of natural numbers regular if it satisfies the following condition: $r_{n+1}/r_n\to \theta \in {\mathbf{R}}^{>1}\cup \{\infty \}$ and, if θ is algebraic, then $(r_n)$ satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of θ. Our main result states that $(\mathbf{Z},+,0,R)$ 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 $(\mathbf{Z},+,0,<,R)$ is NIP whenever R is enumerated by a regular sequence that is ultimately periodic modulo m for all $m>1$.

调用（严格增加）序列 $（{\mathrm{[R}}_{ñ}）$的自然数规律，如果满足以下条件：${\mathrm{[R}}_{ñ+\mathrm{1个}}/{\mathrm{[R}}_{ñ}\to \theta \in {\mathbf{[R}}^{>\mathrm{1个}}\cup \{\infty \}$并且，如果θ是代数的，则$（{\mathrm{[R}}_{ñ}）$满足线性递归关系，其特征多项式为θ的最小多项式。我们的主要结果表明$（\mathbf{ž}，+，0，\mathrm{[R}）$只要R由规则序列枚举，它就是超稳定的。我们给出此结果的两个证明。一个依赖于E. Casanovas和M. Ziegler的结果，另一个依赖于量词消除的结果。我们还表明$（\mathbf{ž}，+，0，<，\mathrm{[R}）$是NIP每当- [R是由有规律序列列举了最终周期性模米为所有$米>\mathrm{1个}$。