SETS DEFINABLE IN ORDERED ABELIAN GROUPS OF FINITE BURDEN (Model theoretic aspects of the notion of independence and dimension)

Abstract

In this note, we survey some recent results on definable sets in ordered Abelian groups of finite burden, focusing on topological and arithmetical tameness properties. In the burden 2 case, and assuming definably completeness, definable discrete subsets of the universe can be characterized as those which are definable in an expansion which is elementarily equivalent to (ℝ;<, +, ℤ). We end with some open questions and possible directions for future research.