Logarithmic--exponential preparation in sharply o-minimal structures

We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\mathbb{R}}_{\exp }^{}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie’s theorem of the complement, a sharp form of Wilkie’s conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion–Rolin. We also develop a parallel complex theory for ${\mathbb{R}}_{\text{an,exp}}$, proving for example that the rational points of height $H$ on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree $\text{poly}(\log H)$. This generalizes a theorem of Cluckers–Pila–Wilkie who proved the same statement for ${\mathbb{R}}_{\text{an}}^{\text{pow}}$.