Wilkie's conjecture for restricted elementary functions

Abstract

We consider the structure ${\mathbb R}^{\mathrm{RE}}$ obtained from $({\mathbb R},<,+,\cdot)$ by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of rational points of height $H$ in the transcendental part of any definable set is bounded by a polynomial in $\log H$. We also prove two refined conjectures due to Pila concerning the density of algebraic points from a fixed number field, or with a fixed algebraic degree, for ${\mathbb R}^{\mathrm{RE}}$-definable sets.