Distal and Non-Distal Pairs

TL;DR

This paper investigates whether certain non-o-minimal, NIP expansions of o-minimal theories are distal, finding that some are distal while others are not, depending on the structure type.

Contribution

It distinguishes between types of non-o-minimal expansions, showing which are distal and which are not, clarifying the boundaries of distality in these theories.

Findings

Tame pairs of o-minimal structures are distal.

Real field with a discrete subgroup is distal.

Dense pairs of o-minimal structures are not distal.

Abstract

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.