On the Pila-Wilkie theorem

TL;DR

This paper provides an accessible overview of the Pila-Wilkie counting theorem, its extensions, and related results, emphasizing simplified proofs and comprehensive treatments of key components in o-minimal geometry.

Contribution

It offers a clear exposition of the Pila-Wilkie theorem, including simplified proofs and detailed explanations of related theorems in o-minimal structures.

Findings

Simplified proof of the Pila-Wilkie theorem using semialgebraic cell decomposition

Complete treatment of Pila and Bombieri's result

Inclusion of the o-minimal Yomdin-Gromov theorem following Binyamini and Novikov

Abstract

This expository paper gives an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin-Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov.