Let G be a multiplicative subgroup of Qp. In this paper, we describe the theory of the pair (Qp, G) under the condition that G satisfies Mann property and is small as subset of a first-order structure. First, we give an axiomatisation of the first-order theory of this structure. This includes an axiomatisation of the theory of the group G as valued group (with the valuation induced on G by the p-adic valuation). If the subgroups G [n] of G have finite index for all n, we describe the definable sets in this theory and prove that it is NIP. Finally, we extend some of our results to the subanalytic setting.

No 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.

Let G be a model of Presburger arithmetic. Let L be an expansion of the language of Presburger L Pres . In this paper, we prove that the L-theory of G is L Pres -minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the L-theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H. Assuming that the induced theories on H and G/H are definable complete and o-minimal respectively, we prove that any definable set of G is L Pres ∪ {H}-definable.