Submission date: 4 July 2017

Abstract:

We consider the four structures (Z; Sqf^Z), (Z; <, Sqf^Z), (Q; Sqf^Q), and (Q; <, Sqf^Q) where Z is the additive group of integers, Sqf^Z is the set of a in Z such that v_p(a) < 2 for every prime p and corresponding p-adic valuation v_p, Q and Sqf^Q are defined likewise for rational numbers, and < denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

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Full text arXiv 1707.00096: pdf, ps.