Iosevich–Rudnev finite-field Falconer distance conjecture

Let $\mathbb{F}_q$ be a finite field of odd cardinality $q$, let $r\geq 2$, and let $E\subseteq\mathbb{F}_q^r$ have cardinality at least $cq^{\frac{r}{2}}$, where $c$ is sufficiently large. Define

$$\Delta(E)=\{\|\mathbf{x}-\mathbf{y}\|:\mathbf{x},\mathbf{y}\in E\},$$

where $\|\mathbf{z}\|=\sum_{i=1}^r z_i^2$. Iosevich–Rudnev conjecture. The distance set satisfies

$$\#\Delta(E)\gg q.$$

This is the finite-field analogue of the Erdős–Falconer distance problem, predicting that sufficiently large subsets contain a positive-proportion number of distinct distances. The source does not provide evidence resolving the conjecture.