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This is a Quantitative Study study.

October 13, 2025Open Access

Sharp o-minimality and lattice point counting

Key Points

The study offers an effective version of the Barroero and Widmer estimates for lattice points in sharp o-minimal structures.
An effective approach is presented for families definable in $ ext{R}_{ ext{exp}}$, enhancing counting results.
Results are grounded in the study of definable families within o-minimal expansions of the real field.
Lattice point counting is crucial for understanding configurations in higher-dimensional spaces.

Abstract

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

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Cite This Study

Harrison-Migochi et al. (Mon,) studied this question.

synapsesocial.com/papers/68ece2abd1bb2827d129743b https://doi.org/https://doi.org/10.48550/arxiv.2503.01731