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Abstract. Let D ⊆ R be closed and discrete and f : D n → R be such that f (D n ) is somewhere dense. We show that (R, +, ·, f) defines Z. As an application, we get that for every α, β ∈ R >0 with log α (β) / ∈ Q, the real field expanded by the two cyclic multiplicative subgroups generated by α and β defines Z.

The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.

Abstract. We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.