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.

The aim of this work is an analysis of distal and non‐distal behavior in dense pairs of o‐minimal structures. A characterization of distal types is given through orthogonality to a generic type in, non‐distality is geometrically analyzed through Keisler measures, and a distal expansion for the case of pairs of ordered vector spaces is computed.

In this note, we construct a distal expansion for the structure $(R, +, <, H)$, where $H \subseteq R$ is a dense $Q$-vector space basis of $R$ (a so-called Hamel basis). Our construction is also an expansion of the dense pair $(R, +, <, Q)$ and has full quantifier elimination in a natural language.