Submission date: 15 February 2017

Abstract:

In [PalSk], Palacin and Sklinos show that the structure (Z,+,0,R) is superstable of U-rank ω when R is either the set of powers of some fixed natural number greater or equal to 2 or a sequence (r_n) of natural numbers such that r_{n+1}/r_n → ∞. In this paper, we generalize this result to the class of sparse sequences (as defined by Semënov) that includes sequences (r_n) given by a recurrence relation for which there exists θ > 1 such that r_n/θ^n has a positive limit and such that the minimal polynomial of θ is the characteristic polynomial of (r_n). We axiomatize for such sequences the structure (Z,+,0,R,S) and prove quantifier elimination in a reasonable language.

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Full text arXiv 1702.04795: pdf, ps.