Sharp Bounds for Rational Points Near Space Curves

Abstract

Let $Q\geq 1$ 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.