Counting algebraic points on definable sets

The counting theorem of Pila and Wilkie (2006) has led to a lively interaction between o-minimality and diophantine geometry in recent years. Part of this interaction has focussed further on the problem of bounding the density of rational and algebraic points lying on transcendental sets, seeking to sharpen the Pila-Wilkie bound in certain cases. We shall survey some results in this area, focussing on a conjecture of Wilkie which concerns the real exponential field. Time permitting, we shall also look at some more recent results (by, amongst others, Besson, Boxall, Jones, Masser, and myself) which concern a variety of classical functions, including the Riemann zeta function, Weierstrass zeta functions and Euler's gamma function

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