Sharp Bounds for Rational Points Near Space Curves

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 $\mathbf{a}\mathrm{/}q$ of height $q\in [1,Q]$ are $\delta \mathrm{/}q$-near $\mathcal{C}$? This manuscript provides an essentially optimal answer, thereby addressing a problem stated by Beresnevich and Kleinbock, for space curves. 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 and Velani.