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.

We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property, regardless of whether it is dense or discrete.

This paper answers several open questions around structures with o‐minimal open core. We construct an expansion of an o‐minimal structure by a unary predicate such that its open core is a proper o‐minimal expansion of. We give an example of a structure that has an o‐minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o‐minimal open core and definable Skolem functions, but is not (...) o‐minimal. (shrink)

Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.

We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o‐minimal structures. Most noteworthily, this property holds for dense pairs of real closed fields, as well as for expansions of o‐minimal structures by a dense independent set.

We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When a is quadratic, addition in the Ostrowski numeration system based on a is recognizable by a finite automaton. We deduce that a subset of X⊆Nn is definable in, where Va is the function that maps a natural number x to the smallest denominator of a convergent of a that appears in the Ostrowski representation based on a of x with a nonzero coefficient if and only if (...) the set of Ostrowski representations of elements of X is recognizable by a finite automaton. The decidability of the theory of follows. (shrink)