Structure theorem for i-minimal expansions of the real additive ordered group

Abstract

We prove that for an o-minimal expansion of the real additive group $\cal R$ and a set $P\subseteq \mathbb{R}$ of dimension $0$ such that $\langle\mathcal{R},P\rangle$ is sparse, has definable choice and every definable set has interior or is nowhere dense then, for every definable set $X$, there is a family $\{X_t:\; t\in A\}$ definable in \Cal R and a set $S\subseteq A$ of dimension $0$ such that $X=\bigcup_{t\in S}X_t$. Moreover, in the d-minimal setting, there is a finite decomposition of $X$ into sets of the previous form such that for every $t\in S$ $X_t$ is relatively open in $\bigcup_{t\in S}X_t$.