Sharp o-minimality and lattice point counting

2025/03/03 by Harrison-Migochi, Andrew, McCulloch, Raymond
#03C64 #03C98 #28A75 #52C07 #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Primary 11H06 #Secondary 11P21

paper · doi:10.48550/arxiv.2503.01731

Abstract

Let Λ⊆ℝⁿ be a lattice and let Z⊆ℝ^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 Z_T=\x∈ℝⁿ:(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_exp.

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