Counting lattice points and o-minimal structures

Abstract

Let $\Lambda$ 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 $Z_T$ to $\Sigma$ 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.

Revised version