The Additive Groups of and With Predicates for Being Square-Free

Abstract

We consider the structures $(\mathbb {Z}; \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Z}; <, \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Q}; \mathrm {SF}^{\mathbb {Q}})$ , and $(\mathbb {Q}; <, \mathrm {SF}^{\mathbb {Q}})$ where $\mathbb {Z}$ is the additive group of integers, $\mathrm {SF}^{\mathbb {Z}}$ 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 $\mathrm {SF}^{\mathbb {Q}}$ 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.

Enriching a predicate and tame expansions of the integers

“…In [KS17], Kaplan and Shelah showed that if P is the set of integers whose absolute value is prime then, assuming a numbertheoretic hypothesis called Dickson's Conjecture, (Z, +, P ) is supersimple of SUrank 1 and unstable. Using a similar strategy, Bhardwaj and Tran [BT21] showed that if S is the set of squarefree integers then (Z, +, S) is supersimple of SU-rank 1 and unstable (without any conditional hypotheses). For both of these results, the proofs involve substantial machinery from number theory (e.g, the proof of instability for (Z, +, P ) uses the work of Green and Tao on arithmetic progressions in primes).…”