2 - Rational points on definable sets

Summary

Abstract

Most of the papers in this volume depend on what has become known as the o-minimal point counting theorem. The aim of this article is to provide enough background in both model theory and number theory for a graduate student in one, but not necessarily both, of these disciplines to be able to understand the statement and the proof of the theorem.

The one dimensional case of the theorem is treated here in full and differs from the original paper ([PW]) in both its number theoretic side (I use the Thue-Siegel Lemma rather than the Bombieri-Pila determinant method) and in its model theoretic side (where the reparametrization of definable functions is made very explicit). The Thue-Siegel method extends easily to the higher dimensional case but, just as in [PW], one has to revert to Yomdin's original inductive argument to extend the reparametrization, and this is only sketched here.

I am extremely grateful to Adam Gutter for typing up my handwritten notes of the Manchester LMS course on which this paper is based. Also, my deepest thanks to Margaret Thomas for so carefully reading the original manuscript. Her numerous suggestions have greatly improved the presentation.

Introduction

So, the aim of these notes is to prove the following:

1.1 Theorem (Pila-Wilkie [PW])

Let S ⊆ Rⁿ be a set definable in some o-minimal expansion of the ordered field of real numbers. Assume that S contains no infinite, semi-algebraic subset. Let ∊ > 0 be given. Then for all sufficiently large H, the set S contains at most H^∊ rational points of height at most H.

• The underlined terms will be defined below.