Diophantine approximation with the determinant method

The celebrated theorem of Pila and Wilkie, giving a subpolynomial bound for the number of rational points of bounded height on certain transcendental sets, has had far-reaching consequences in number theory. Recent work of Habegger gives a parallel result for points which are only close to a transcendental set. We discuss possible extensions of this result and their number-theoretic implications. This talk is based on joint work with Ryan Keast and Margaret Thomas.