AI 中文总结
该论文在离散有限域场景改进了单纯形分布的指数下界，在连续欧氏场景针对紧集与紧Salem集证明了带标记单纯形锚定距离配置测度的绝对连续性，完善了相关几何测度论问题的研究。
AI 中文摘要
本文研究离散与连续场景下单纯形的分布。设$q$为奇素数幂，$Q$为$\boldsymbol{F}_q^d$上的非退化二次型，$2 \leq k \leq d-1$，我们证明所有满足$|E| \backslashgeq C_{d,k}q^{\beta_{d,k}}$（其中$\beta_{d,k}$的表达式为：当$d-k$为偶数时，$\beta_{d,k}=\frac{d+k}{2}-\frac{k-1}{k+1}$；当$d-k$为奇数时，$\beta_{d,k}=\frac{d+k-1}{2}$）的集合$E \backslashsubset \boldsymbol{F}_q^d$，可确定所有有序非退化$k$-单纯形同余类的正比例，该结果改进了Bennett、Hart、Iosevich、Pakianathan与Rudnev（2017）此前给出的指数，且在$d-k$为奇数时是最优的。在欧氏场景下，我们证明若$E \backslashsubset \boldsymbol{R}^d$为紧集且$\text{dim}_\text{H}(E)>d-1$，则存在支撑于$E$上的Frostman概率测度$\boldsymbol{\text{μ}}$，以及具有全$\boldsymbol{\text{μ}}$测度的锚点集，使得带标记$(d-1)$-单纯形的锚定距离配置测度在每个此类锚点处均为绝对连续；还证明当$E \backslashsubset \boldsymbol{R}^d$为满足$\text{dim}_\text{H}(E)>k$的紧Salem集时，上述结论同样成立。
英文摘要
In this paper, we study the distribution of simplices in both discrete and continuous settings. Let $q$ be an odd prime power, let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^d$, and let $2\leq k\leq d-1$. We prove that every set $E\subset\mathbb F_q^d$ with \[ |E|\geq C_{d,k}q^{β_{d,k}}, \qquad β_{d,k}= \begin{cases} \displaystyle \frac{d+k}{2}-\frac{k-1}{k+1}, & d-k\ \text{even},\\[2mm] \displaystyle \frac{d+k-1}{2}, & d-k\ \text{odd}, \end{cases} \] determines a positive proportion of all ordered nondegenerate $k$-simplex congruence classes. This improves the previous exponent due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017), and is sharp when $d-k$ is odd. In the Euclidean setting, we prove that if $E\subset\mathbb R^d$ is compact and $\dim_{\mathrm H}(E)>d-1$, then there exists a Frostman probability measure $μ$, supported on $E$, and a set of pins of full $μ$-measure such that the pinned distance configuration measure for labeled $(d-1)$-simplices is absolutely continuous at every such pin. We also show that the same conclusion holds when $E\subset\mathbb R^d$ is a compact Salem set with $\dim_{\mathrm H}(E)>k$.
Comments44 pages
