The Pila-Wilkie Theorem and variations

Pila and Wilkie's influential counting theorem provides a bound on the density of rational points of bounded height lying on the 'transcendental parts' of sets definable in o-minimal expansions of the real field. This result has brought about a lively interaction in recent years between o-minimality and diophantine geometry, including several important applications to arithmetical conjectures which will be explored further in Ya'acov Peterzil's tutorial. As a prelude to this, we will provide an introduction to the Pila-Wilkie Theorem, indicating the main ingredients involved in the proof. In particular, we will focus on the key step known as the Pila-Wilkie Reparameterization Theorem. This is a model theoretic statement about the geometry of sets definable in o-minimal expansions of real closed fields - namely that they can be covered by the images of finitely many sufficiently differentiable functions with bounded derivatives. Following the Pila-Wilkie Theorem, subsequent work carried out by a number of authors, including Pila, Besson, Boxall, Butler, Jones, Masser and myself, has focussed on establishing that a sharper bound holds in certain situations. One important goal is a conjecture of Wilkie concerning sets definable in the real exponential field. We shall explore some of the cases of this conjecture already established and the methods involved, indicating how a suitable modification of the Pila-Wilkie notion of parameterization could play an important role in the pursuit of this conjecture.

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