标题： 加性结构在$\mathbb{F}_q^d$中意味着更多的距离 显示英文标题

标题： Additive structures imply more distances in $\mathbb{F}_q^d$

摘要： 对于集合$E \subseteq \mathbb{F}_q^d$，距离集定义为$\Delta(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$，其中$\|\cdot\|$表示标准二次型。我们研究了在Fraser引入的灵活类$(u, s)$-Salem集中的Erdős--Falconer距离问题，重点在于偶数情况$u = 4$。 通过利用$\|\widehat{E}\|_4$与第四阶加性能量$\Lambda_4(E)$之间的精确恒等式，我们证明了$\Lambda_4(E)$的数量化改进会迫使存在许多距离。 特别是，对于一个$(4, s)$-Salem 集$E\subset \mathbb{F}_q^d$，当$d \geq 2$时，我们证明如果\[ |E|\gg q^{\min\left\{\frac{d+2}{4s+1}, \frac{d+4}{8s}\right\}}, \]，则$E$确定所有距离的正比例。 这严格改进了 Fraser 在某些参数范围内的阈值$\frac{d}{4s}$和 Iosevich-Rudnev 的界限$q^{\frac{d+1}{2}}$。 作为应用，我们获得了任意代数簇上乘法子群和集合的改进阈值，并建立了在独立于相交几何中的尖锐相交界。 我们还提出了一个统一的猜想，针对$(4, s)$-Salem 集，该猜想调和了已知的界限并指明了奇维球面区域：在奇维数$d \geq 3$中，经常引用的$\frac{d-1}{2}$门槛在没有额外结构的情况下并不成立，而在原始半径球面上，任何$q^{-\epsilon/2}$的增益在第四能量中都能提高$\frac{d}{2}$的现有门槛。 这提供了一种新方法来解决这个问题。 显示英文摘要

摘要： For a set $E \subseteq \mathbb{F}_q^d$, the distance set is defined as $\Delta(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$, where $\|\cdot\|$ denotes the standard quadratic form. We investigate the Erd\H{o}s--Falconer distance problem within the flexible class of $(u, s)$--Salem sets introduced by Fraser, with emphasis on the even case $u = 4$. By exploiting the exact identity between $\|\widehat{E}\|_4$ and the fourth additive energy $\Lambda_4(E)$, we prove that quantitative gains in $\Lambda_4(E)$ force the existence of many distances. In particular, for a $(4, s)$--Salem set $E\subset \mathbb{F}_q^d$ with $d \geq 2$, we prove that if \[ |E|\gg q^{\min\left\{\frac{d+2}{4s+1}, \frac{d+4}{8s}\right\}}, \] then $E$ determines a positive proportion of all distances. This strictly improves Fraser's threshold of $\frac{d}{4s}$ and the Iosevich-Rudnev bound of $q^{\frac{d+1}{2}}$ in certain parameter ranges. As applications, we obtain improved thresholds for multiplicative subgroups and sets on arbitrary varieties, and establish a sharp incidence bound for Salem sets that is of independent interest in incidence geometry. We also propose a unified conjecture for $(4, s)$--Salem sets that reconciles known bounds and pinpoints the odd-dimensional sphere regime: in odd dimensions $d \geq 3$, the often-cited $\frac{d-1}{2}$ threshold does not follow without additional structure, while on primitive-radius spheres, any $q^{-\epsilon/2}$-gain in the fourth energy improves the standing threshold of $\frac{d}{2}$. This provides a new approach to address this problem.