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

We consider the four structures $(\mathbb{Z};{\mathrm{S}\mathrm{q}\mathrm{f}}^{\mathbb{Z}})$, $(\mathbb{Z};<,{\mathrm{S}\mathrm{q}\mathrm{f}}^{\mathbb{Z}})$, $($\mathbb{Q}$; $\mathrm{S}\mathrm{q}\mathrm{f}$^$\mathbb{Q}$)$,and$($\mathbb{Q}$; <, $\mathrm{S}\mathrm{q}\mathrm{f}$^$\mathbb{Q}$)$where$$\mathbb{Z}$$istheadditivegroupofintegers,$$\mathrm{S}\mathrm{q}\mathrm{f}$^$\mathbb{Z}$$ is the set of $a\in \mathbb{Z}$ such that $v_p(a)<2$ for every prime $p$ and corresponding $p$-adic valuation $v_p$, $\mathbb{Q}$ and ${\mathrm{S}\mathrm{q}\mathrm{f}}^{\mathbb{Q}}$ 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.