Sharp o-minimality and lattice point counting

Let $\mathrm{\Lambda }\subseteq {\mathbb{R}}^n$ be a lattice and let $Z\subseteq {\mathbb{R}}^{m+n}$ be a definable family in an o-minimal expansion of the real field, $\bar{\mathbb{R}}$. A result of Barroero and Widmer gives sharp estimates for the number of lattice points in the fibers $Z_T=\{x\in {\mathbb{R}}^n:(T,x)\in Z\}$. Here we give an effective version of this result for a family definable in a sharply o-minimal structure expanding $\bar{\mathbb{R}}$. We also give an effective version of the Barroero and Widmer statement for certain sets definable in ${\mathbb{R}}_{\exp }$.