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

TL;DR

This paper extends the Pila-Wilkie theorem to o-minimal structures expanded by a dense set, showing that definable sets with many rational points are dense in infinite semialgebraic sets and contain definable infinite subsets.

Contribution

It generalizes the Pila-Wilkie theorem to expansions of o-minimal structures by dense sets, including elementary substructures and independent sets.

Findings

Definable sets with many rational points are dense in infinite semialgebraic sets.

Such sets contain infinite definable subsets in the expanded structure.

The results apply to expansions by dense elementary substructures or independent sets.

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 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.