Sharp Bounds for Rational Points Near Space Curves

Abstract

Let $Q\ge 1$ be large, and $\delta \in (0,1)$ be small. Denote by $\mathcal{C}\subset {\mathbb{R}}^3$ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points $\mathbf{a}/q$ of height $q\in [Q,2Q]$ are $\delta /q$-near $\mathcal{C}$? This manuscript provides an essentially optimal answer. We show that the folklore conjectures are incorrect for certain manifolds with codimension $\ge 2$, including the moment curve $(t,t^2,t^3)$. The reason is a hitherto hidden `major arc' type obstruction. We also establish matching upper bounds (up to endpoints). Our argument combines purely Fourier analytical techniques with the planar counting results by Vaughan--Velani.

49 pages, including glossary. Comments welcome