Distribution of simplices in the discrete and continuous settings

Abstract

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 \[ |E|\geq C_{d,k}q^{\beta_{d,k}}, \qquad \beta_{d,k}= \begin{cases} \displaystyle \frac{d+k}{2}-\frac{k-1}{k+1}, & d-k\ \text{even},\\[2mm] \displaystyle \frac{d+k-1}{2}, & d-k\ \text{odd}, \end{cases} \] 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 $\mu$, supported on $E$, and a set of pins of full $\mu$-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 $\dim_{\mathrm H}(E)>k$.