Sharp o-minimality and lattice point counting
Andrew Harrison-Migochi, Raymond McCulloch
TL;DR
This work provides an effective lattice-point counting framework for sets definable in sharp o-minimal structures, extending Barroero–Widmer's results to a #o-minimal context and to certain ℝ_exp-definable families. By introducing FD-filtrations and leveraging sharp cell-decomposition, the authors obtain explicit polynomial-in-degree bounds on lattice-point deviations, with the key bound |Z_T ∩ Λ| − Vol(Z_T)/det(Λ)| ≤ poly(F)(D) ∑_{j=0}^{n−1} V_j(Z_T)/(λ_1…λ_j). They adapt Davenport-type arguments to definable families, prove uniform bounds for projections and boundary contributions, and extend the theory to existential ℝ_exp-definable sets via restricted sub-Pfaffian machinery. The results yield effective, computable constants depending only on the definable family’s format and degree, enabling explicit lattice-counting in broad definable classes and offering a path to effective Wilkie-type results in ℝ_exp. This advances the interface between model theory, geometry of numbers, and Diophantine counting in structured nonlinear settings.
Abstract
Let $Λ\subseteq\mathbb{R}^n$ be a lattice and let $Z\subseteq\mathbb{R}^{m+n}$ be a definable family in an o-minimal expansion of the real field, $\overline{\mathbb{R}}$. 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 $\overline{\mathbb{R}}$. We also give an effective version of the Barroero and Widmer statement for certain sets definable in $\mathbb{R}_{\exp}$.