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arXiv · 2610.06422

How close can rational points get to a manifold?

Victor Beresnevich·Shreyasi Datta
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Abstract

We prove the heuristically predicted lower bound for the number of rational points of height at most $Q$ lying within $ε/Q$ of a fixed analytic nondegenerate manifold in $\mathbb{R}^n$, provided that $ε\asymp Q^{-τ}$ for some $τ\leq 3/(2m+1)$, where $m$ is the codimension of the manifold. Our result establishes the lower bound well beyond the previously conjectured range $τ\leq 1/m$, and improves upon a recent result of Schindler, Srivastava, and Technau, who established the same lower bound for $τ\leq 3/(2n-1)$.

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Victor Beresnevich, Shreyasi Datta. 2026-10-05. How close can rational points get to a manifold?. https://arxiv.org/abs/2610.06422

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