发表机构
University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对短区间的指数和估计，改进了结合经典Weyl差分与最优Vinogradov均值定理的通用界，得到了更优的Weyl界，完善了短区间上的指数和估计理论。
AI 中文摘要
对于整数 $d\ge 3$，令 $\Delta_d=\min\{2^{d-1},d(d-1)\}$；设 $a/q$ 为既约分数，$P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$，$\mathcal{I}$ 是长度 $H\le q$ 的连续整数区间。我们证明 $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\Delta_d+\varepsilon}$。由此，对每个素数 $p>d$、每个 $\mathbb{F}_p[X]$ 中 $d$ 次多项式 $P$，以及每个满足 $p^{1/d}<H<p^{1/(d-1)}$ 的长度为 $H$ 的连续整数区间 $\mathcal{I}$，记 $H^d/p=H^u$，可得 $\left|\sum_{n\in\mathcal{I}}e_p(P(n))\right|\ll_{d,\varepsilon}H^{1-\min\{u/d,1/\Delta_d\}+\varepsilon}$，该结果在整个自然短区间范围内，严格改进了结合经典Weyl差分与最优Vinogradov均值定理得到的最佳通用估计。
英文摘要
For an integer $d\ge 3$, put $Δ_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+α_{d-1}X^{d-1}+\cdots+α_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/Δ_d+\varepsilon}$. Consequently, for every prime $p>d$, every degree-$d$ polynomial $P\in\mathbb{F}p[X]$, and every interval $\mathcal{I}$ of $H$ consecutive integers with $p^{1/d}<H<p^{1/(d-1)}$, writing $H^d/p=H^u$, one has $\left|\sum{n\in\mathcal{I}}e_p(P(n))\right|\ll_{d,\varepsilon}H^{1-\min{u/d,1/Δ_d}+\varepsilon}$. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.
Comments19 pages, no figures. Comments are welcome. Formalization project: https://github.com/hxypqr/improved-weyl-bounds-short-intervals
