Counting lattice points and o-minimal structures

TL;DR

This paper provides sharp estimates for counting lattice points within definable families in o-minimal structures, linking geometric projections to lattice point enumeration with bounds depending only on the family.

Contribution

It introduces precise bounds for lattice point counts in o-minimal definable sets, connecting projections' volumes to lattice point enumeration.

Findings

Sharp estimates for lattice points in definable families

Bound on projected volumes related to coordinate subspaces

Results depend only on the definable family

Abstract

Let $\Lambda$ be a lattice in $\R^n$, and let $Z\subseteq \R^{m+n}$ be a definable family in an o-minimal structure over $\R$. We give sharp estimates for the number of lattice points in the fibers $Z_T={x\in \R^n: (T,x)\in Z}$. Along the way we show that for any subspace $\Sigma\subseteq\R^n$ 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.