The finite-field Erdős distance conjecture

Let ${\mathbb F}_q$ be a finite field with $q$ elements, let ${\mathbb F}_q^d$ be its $d$-dimensional vector space, and let $E\subset {\mathbb F}_q^d$ with $d\geq 2$. Define

$$\Delta(E)=\{(x_1-y_1)^2+\dots+(x_d-y_d)^2:x,y\in E\}\subset {\mathbb F}_q.$$

Write $X\lesssim Y$ when $X\leq CY$ for some constant $C>0$, and $X\gtrsim Y$ when $Y\lesssim X$.

Finite-field Erdős distance conjecture. If

$$\#E\lesssim q^{d/2},$$

then

$$\#\Delta(E)\gtrsim (\#E)^{2/d}.$$

The unrestricted Euclidean-type estimate fails for large subsets, such as $E={\mathbb F}_q^d$, for which $\#E=q^d$ but $\#\Delta(E)=q$. The conjecture is presented as open in the source and is the finite-field analogue of the Falconer distance problem.