Topological properties of definable sets in ordered Abelian groups of burden 2

TL;DR

This paper investigates the topological structure of definable sets in ordered Abelian groups of burden 2, revealing restrictions on their complexity and providing explicit examples of such groups with rich definable sets.

Contribution

It establishes new topological constraints for definable sets in burden 2 ordered Abelian groups and constructs explicit examples illustrating these properties.

Findings

Infinite discrete definable sets prevent certain dense-codense sets from existing.

In burden 2 structures, definable dense-codense sets imply the Independence Property.

An explicit example of a burden 2 ordered Abelian group with both discrete and dense-codense definable sets.

Abstract

We obtain some new results on the topology of unary definable sets in densely ordered Abelian groups of burden groups of burden 2. In the special case in which the structure has dp-rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle-third set. If the structure has burden 2 and both an infinite discrete set D and a dense-codense set X are definable, then translates of X must witness the Independence Property. In the last section, an explicit example of an ordered Abelian group of burden 2 is given in which both an infinite discrete set and a dense-codense set are definable.