Sharp o-minimality and lattice point counting

Let Λⁿ be a lattice and let Zᵐ⁺ⁿ be a definable family in an o-minimal expansion of the real field, R̄. A result of Barroero and Widmer gives sharp estimates for the number of lattice points in the fibers ZT=ⁿ:(T,x)∈ Z\. Here we give an effective version of this result for a family definable in a sharply o-minimal structure expanding R̄. We also give an effective version of the Barroero and Widmer statement for certain sets definable in Rₑₓₚ.