We show that if a first-order structure${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then${\cal M}$must be interdefinable with (ℤ,+,0) or (ℤ,+,<,0).

Call a (strictly increasing) sequence (rn) of natural numbers regular if it satisfies the following condition: rn+1/rn→θ∈R>1∪{∞} and, if θ is algebraic, then (rn) satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of θ. Our main result states that (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 (Z,+,0,<,R) is NIP whenever R is enumerated by a regular sequence that is ultimately periodic modulo m for all m>1. (shrink)

We prove that has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank ω. Additionally, our methods yield other superstable expansions such as equipped with the set of factorial elements.