Nonvaluational ordered Abelian groups of finite burden

Abstract

Consider an expansion $\mathcal{R}=(R,<,+,\dots )$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $R$ which is dense and codense in a definable open subset $U$ of $R$ with $X\subseteq U$. We further assume that $\mathcal{R}$ is nonvaluational, that is, for every nonempty definable subsets $A,B$ of $R$ with $A<B$ and $A\cup B=R$, $\inf \{b-a|a\in A,b\in B\}=0$. Then, $\mathcal{R}$ is $\ast$-locally weakly o-minimal. We also give a complete description of sets definable in a definably complete expansion of ordered group of burden two if it defines an infinite discrete set.