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.