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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 everydefinable 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, inthe d-minimal setting, there is a finite decomposition of $X$ into sets of theprevious form such that for every $t\in S$ $X_t$ is relatively open in$\bigcup_{t\in S}X_t$.

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