Quantitative Oppenheim Conjecture for Random Quadratic Forms and Optimal Variance Bounds in Function Fields

Jiyoung Han; Noy Soffer Aranov

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arXiv · 2606.16699

Quantitative Oppenheim Conjecture for Random Quadratic Forms and Optimal Variance Bounds in Function Fields

Jiyoung Han·Noy Soffer Aranov

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Abstract

We prove a quantitative version of Oppenheim's conjecture in the function field setting. In order to do so, we compute the higher moments of the Siegel transform. In particular, we find an optimal bound on the variance of the number of lattice points in a set. Moreover, we compute the exact variance of the number of lattice points in a ball, which is of independent interest.

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Jiyoung Han, Noy Soffer Aranov. 2026-06-15. Quantitative Oppenheim Conjecture for Random Quadratic Forms and Optimal Variance Bounds in Function Fields. https://arxiv.org/abs/2606.16699

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