发表机构
Institut des Hautes Études Scientifiques, Université Paris-Saclay; University of Wisconsin（高等科学研究所，巴黎萨克雷大学; 威斯康星大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究空间曲线附近高度在[Q,2Q]内的有理点计数问题，结合傅里叶分析与Vaughan–Velani平面计数结果，证明余维数≥2的部分流形的民间猜想不成立，给出了相关精确界。
AI 中文摘要
设Q≥1为大数，δ∈(0,1)为小数，记ℝ³中具有非零曲率与挠率的足够光滑曲线为𝒞，求高度q∈[Q,2Q]的有理点𝐚/q中，与𝒞的距离为δ/q的点的数量。本文给出了本质最优的答案：证明了余维数≥2的某些流形（包括矩曲线(t,t²,t³)）的民间猜想不成立，原因是存在此前未被发现的“主弧”型障碍；还建立了匹配的上界（端点除外），论证结合了纯傅里叶分析技术与Vaughan–Velani的平面计数结果。
英文摘要
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 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.
Comments52 pages, including glossary. Comments welcome. Version 2 includes applications of our main theorem
