Abstract:
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.
