Counterexamples for Rational Points Near Curves

Let $δ\in [0,1/2]$ and $Q\geq 1$. Given a compact $C^{\infty}$-curve $\mathcal{C}$ in $\mathbb{R}^n$, denote by $N_{\mathcal{C}}(δ, Q)$ the number of pairs $(\mathbf{a}, q)\in \mathbb{Z}^n\times [Q/2,Q]$ such that the rational point $\mathbf{a}/q$ is $δ/q$-close to $\mathcal{C}$. We investigate the