Sharp Bounds for Rational Points Near Space Curves

Let $Q\geq 1$ be large, and $δ\in(0,1)$ 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 $δ/q$-near $\mathcal C$? This manuscript provides an essentially optim