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arXiv · 2512.23204

Counting rational points near manifolds: a refined estimate, a conjecture and a variant

Jonathan Hickman·Rajula Srivastava·James Wright
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Abstract

Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$ case we formulate a conjecture concerning this count. The conjecture is motivated in part by interpreting certain codimension $2$ submanifolds of $\mathbb{R}^{2m+2}$ as complex hypersurfaces in $\mathbb{C}^{m+1}$ and using the complex structure to provide a natural reformulation of the curvature condition. Finally, we provide further evidence for the conjecture by proving a natural variant for $n \geq 2$ in which rationals are replaced with Gaussian rationals.

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Jonathan Hickman, Rajula Srivastava, James Wright. 2025-12-29. Counting rational points near manifolds: a refined estimate, a conjecture and a variant. https://arxiv.org/abs/2512.23204

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