Counting Lattice Points and O-Minimal Structures

Let Λ be a lattice in $\mathbb R^n$, and let $Z\subseteq\mathbb R^{m+n}$ be a definable family in an O-minimal structure over $\mathbb R$. We give sharp estimates for the number of lattice points in the fibers $Z_T=\{x\in\mathbb R^n:(T, x)\in Z\}$. Along the way, we show that for any subspace $\Sigma \subseteq \mathbb R^n$ of dimension j>0 the j-volume of the orthogonal projection of ZT to Σ is, up to a constant depending only on the family Z, bounded by the maximal j-dimensional volume of the orthogonal projections to the j-dimensional coordinate subspaces.