Counting Algebraic Points in Expansions of O-Minimal Structures by a Dense Set

Abstract

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 $\widetilde{\mathcal{R}}=\langle {\mathcal{R}}, P\rangle$ of ${\mathcal{R}}$ by a dense set P, which is either an elementary substructure of ${\mathcal{R}}$, or it is $\mathrm{dcl}$-independent, as follows. If X is definable in $\widetilde{\mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ${\emptyset}$-definable in $\langle \overline{\mathbb{R}}, P\rangle$, where $\overline{\mathbb{R}}$ is the real field. Along the way we introduce the notion of the ‘algebraic trace part’ $X^{{\, alg}}_t$ of any set $X\subseteq \mathbb{R}^n$, and we show that if X is definable in an o-minimal structure, then $X^{{\, alg}}_t$ coincides with the usual algebraic part of X.