Counting rational points on transcendental sets

Let $\mathrm{X}$ be a set definable in some o-minimal structure, for example a real analytic subset of $\mathbb{R}^n$. The Pila–Wilkie theorem (in its basic form) states that the number of rational points in the transcendental part of X grows sub-polynomially with the height of the points. The Wilkie conjecture stipulates that for sets definable in $R_{\exp}$, one can sharpen this asymptotic to polylogarithmic. I will describe a complex-analytic approach to the proof of the Pila–Wilkie theorem for subanalytic sets. I will then discuss how this approach leads to a proof of the "restricted Wilkie conjecture", where we replace $R_{\exp}$ by the structure generated by the restrictions of $\exp$ and $\sin$ to the unit interval.
Joint work with Gal Binyamini.