Wilkie's conjecture for restricted elementary functions

Abstract

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