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

Abstract

We establish sharp bounds for rational points near compact smooth nondegenerate curves in $\mathbb R^n$, for every $n\geq4$. For a nondegenerate curve $\mathcal C$ in $\mathbb R^n$, let \[ \mathcal R_{\mathcal C}(δ,Q) :=\#\left\{(\mathbf p,q)\in\mathbb Z^n\times\mathbb N: 1\leq q\leq Q,\ \operatorname{dist}(\mathbf p/q,\mathcal C)\leqδ/q\right\}.\] We prove that \[\mathcal R_{\mathcal C}(δ,Q) \lesssim_{\varepsilon,\mathcal C} δ^{n-1}Q^2+Q^\varepsilon \left(Q+\sum_{k=1}^{n-1} δ^{k^2/(k+1)}Q^{(2k+1)/(k+1)}\right).\] A key new ingredient is a wave packet method developed by Gan--Maldague--Oh.