Call a (strictly increasing) sequence R = (rn) of natural numbers regular if it satisfies the following condition: r n+1 /rn → θ ∈ R >1 ∪ {∞} and, if θ is algebraic, then R satisfies a recurrence relation whose characteristic polynomial is the minimal polynomial of θ.Our main result states that Z R = (Z, +, 0, R) is superstable whenever R is a regular sequence. We provide two proofs of this result. One relies on a result of E. Casanovas and M. Ziegler and the other on a quantifier elimination result. Both proofs share an important ingredient: for a regular sequence, the set of solutions of homogeneous equations is either finite or controlled by (finitely many) recurrence relations. This property resembles the Mann property defined by L. van den Dries and A. Günaydın in their work on expansions of fields by subgroups. Inspired by their work and the proofs of our main result, we show that when M is the domain of a multiplicative monoid of integers with the Mann property, then Z M is also superstable.
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