Sharp rational points counting near nondegenerate curves in ${\mathbb{R}}^n$

We establish sharp bounds for rational points near compact smooth nondegenerate curves in ${\mathbb{R}}^n$, for every $n\ge 4$. For a nondegenerate curve $\mathcal{C}$ in ${\mathbb{R}}^n$, let $${\mathcal{R}}_{\mathcal{C}}(\delta ,Q):=\mathrm{\#}\{(\mathbf{p},q)\in {\mathbb{Z}}^n\times \mathbb{N}:1\le q\le Q,\mathrm{dist}(\mathbf{p}\mathrm{/}q,\mathcal{C})\le \delta \mathrm{/}q\}\mathrm{.}$$ We prove that $${\mathcal{R}}_{\mathcal{C}}(\delta ,Q){\lesssim }_{\epsilon ,\mathcal{C}}{\delta }^{n-1}{Q}^2+Q^{\epsilon }(Q+\sum _{k=1}^{n-1}{\delta }^{k^2\mathrm{/}(k+1)}{Q}^{(2k+1)\mathrm{/}(k+1)})\mathrm{.}$$ A key new ingredient is a wave packet method developed by Gan–Maldague–Oh.