Counting algebraic points in expansions of o-minimal structures by a dense set

Abstract

The Pila-Wilkie theorem states that if a set $X\subseteq {\mathbb{R}}^n$ is definable in an o-minimal structure $\mathcal{R}$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\tilde{\mathcal{R}}=⟨\mathcal{R},P⟩$ of $\mathcal{R}$ by a dense set $P$, which is either an elementary substructure of $\mathcal{R}$, or it is independent, as follows. If $X$ is definable in $\tilde{\mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\varnothing$-definable in $⟨\overline{\mathbb{R}},P⟩$, where $\overline{\mathbb{R}}$ is the real field.