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 $q\in [1,Q]$ are $\delta /q$-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.

52 pages, including glossary. Comments welcome. Version 2 includes applications of our main theorem