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We consider the four structures $(\mathbb{Z}; \mathrm{Sqf}^\mathbb{Z})$,$(\mathbb{Z}; <, \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Q};\mathrm{Sqf}^\mathbb{Q})$, and $(\mathbb{Q}; <, \mathrm{Sqf}^\mathbb{Q})$ where$\mathbb{Z}$ is the additive group of integers, $\mathrm{Sqf}^\mathbb{Z}$ isthe set of $a \in \mathbb{Z}$ such that $v_{p}(a) < 2$ for every prime $p$ andcorresponding $p$-adic valuation $v_{p}$, $\mathbb{Q}$ and$\mathrm{Sqf}^\mathbb{Q}$ are defined likewise for rational numbers, and $<$denotes the natural ordering on each of these domains. We prove that the secondstructure is model-theoretically wild while the other three structures aremodel-theoretically tame. Moreover, all these results can be seen as exampleswhere number-theoretic randomness yields model-theoretic consequences.

The additive groups of $\mathbb{Z}$ and $\mathbb{Q}$ with predicates for being square-free