Expansions of the p‐adic numbers that interpret the ring of integers

Abstract

Let truedouble-struckQp∼ be the field of p‐adic numbers in the language of rings. In this paper we consider the theory of truedouble-struckQp∼ expanded by two predicates interpreted by multiplicative subgroups αZ and βZ where α,β∈N are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p‐adic valuation. If either α or β has zero valuation we show that the theory of (Qp∼,αdouble-struckZ,βdouble-struckZ) has the NIP (“negation of the independence property”) and therefore does not interpret Peano arithmetic. In that case we also prove that the theory is decidable if and only if the theory of (Qp∼,αdouble-struckZβdouble-struckZ) is decidable.