发表机构
Institut des Hautes Études Scientifiques; University of Wisconsin（高等科学研究所; 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
本文针对矩曲线附近有理点计数问题，证明已知的指数范围在渐近意义下接近最优，至多只能改进 $O(n^{-3})$，否定了此前 $O(n^{-1})$ 改进的普遍预期。
AI 中文摘要
设 $\delta \in [0,1/2]$ 且 $Q\geq 1$。给定 $\mathbb{R}^n$ 中的紧致 $C^{\infty}$ 曲线 $\mathcal{C}$，记 $N_{\mathcal{C}}(\delta, Q)$ 为满足有理点 $\mathbf{a}/q$ 与 $\mathcal{C}$ 的距离不超过 $\delta/q$ 的配对 $(\mathbf{a}, q)\in \mathbb{Z}^n\times [Q/2,Q]$ 的数量。我们研究使得启发式估计 $N_{\mathcal{C}}(\delta, Q)\asymp\delta^{n-1}Q^2$ 对矩曲线成立所必需的 $\delta$ 关于 $Q$ 的范围。对于非退化曲线 $\mathcal{C}\subset \mathbb{R}^n$ 且 $Q$ 足够大时，Hickman 和第二作者先前已证明对所有 $\delta\in[Q^{\alpha(n+1)+\nu},1/2)$ 及任意 $\nu>0$，有 $N_{\mathcal{C}}(\delta, Q)\lesssim_{\nu} \delta^{n-1}Q^2$。此处 $\alpha(n)=-4n^{-2}+O(n^{-3})$ 已被显式计算。我们证明在 $n\to\infty$ 的渐近意义下，该范围惊人地接近最优，且至多可扩展至 $\delta\in[Q^{\alpha(n)},1/2)$。特别地，最优指数至多只能改进约 $O(n^{-3})$，尽管此前普遍认为可能改进 $O(n^{-1})$。剩余范围内的互补上界最近由 Gan--Guo--Oh 建立。
英文摘要
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 range of $δ$ in terms of $Q$ which is necessary for the heuristic $N_{\mathcal{C}}(δ, Q)\asympδ^{n-1}Q^2$ to be correct for the moment curve. For a nondegenerate curve $\mathcal{C}\subset \mathbb{R}^n$ and for sufficiently large $Q$, Hickman and the second author previously established that $N_{\mathcal{C}}(δ, Q)\lesssim_ν δ^{n-1}Q^2$ for all $δ\in[Q^{α(n+1)+ν},1/2)$ and any $ν>0$. Here $α(n)=-4n^{-2}+O(n^{-3})$ was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as $n\to\infty$, and can at most be extended to $δ\in[Q^{α(n)},1/2)$. In particular, the sharp exponent can only be about $O(n^{-3})$ better, even though it was widely believed before that an improvement of $O(n^{-1})$ should be possible. The complementary upper bound in this remaining range has been recently established by Gan--Guo--Oh.
Comments5 pages, comments welcome
