Expansions of the $p$-adic numbers that interprets the ring of integers

Abstract

Let $\widetilde{{\mathbb{Q}}_p}$ be the field of $p$-adic numbers in the language of rings. In this paper we consider the theory of $\widetilde{{\mathbb{Q}}_p}$ expanded by two predicates interpreted by multiplicative subgroups ${\alpha }^{\mathbb{Z}}$ and ${\beta }^{\mathbb{Z}}$ where $\alpha ,\beta \in \mathbb{N}$ are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if $\alpha$ and $\beta$ have positive $p$-adic valuation. If either $\alpha$ or $\beta$ has zero valuation we show that the theory of $(\widetilde{{\mathbb{Q}}_p},{\alpha }^{\mathbb{Z}},{\beta }^{\mathbb{Z}})$ does not interpret Peano arithmetic. In that case we also prove that the theory is decidable iff the theory of $(\widetilde{{\mathbb{Q}}_p},{\alpha }^{\mathbb{Z}}\cdot {\beta }^{\mathbb{Z}})$ is decidable.