Externally definable quotients and NIP expansions of the real ordered additive group

Abstract: Let $\mathcal{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathcal{R}$ is generically locally o-minimal. It follows that if $X \subseteq \mathbb{R}^n$ is definable in $\mathcal{R}$ then the $C^k$-points of $X$ are dense in $X$ for any $k \geq 0$. This follows from a more general theorem on $\mathrm{NIP}$ expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable and $\bigwedge$-definable. We also show that $\mathcal{R}$ is strongly dependent if and only if $\mathcal{R}$ is either o-minimal or $(\mathbb{R},<,+,\alpha\mathbb{Z})$-minimal for some $\alpha > 0$.