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

## TL;DR

This paper studies expansions of the $p$-adic numbers by multiplicative subgroups, showing that certain expansions interpret Peano arithmetic and analyzing their decidability based on valuations of the subgroups.

## Contribution

It introduces a new framework for expanding $p$-adic numbers with multiplicative subgroups and characterizes when these expansions interpret arithmetic and are decidable.

## Key findings

- Interpretation of Peano arithmetic depends on valuations of subgroups.
- Decidability of the theory relates to the combined subgroup structure.
- The theory is undecidable when both subgroups have positive valuation.

## 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.