The finite-field Erdős–Falconer distance conjecture in even dimensions

Let $q$ be a prime power, let $d\geq 2$ be even, and let $E\subset\mathbb{F}_q^d$. Define the distance set

$$\Delta(E)=\left\{\|x-y\|:=\sum_{i=1}^d(x_i-y_i)^2:x,y\in E\right\}.$$

Erdős–Falconer distance conjecture. If $|E|\geq Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, then

$$|\Delta(E)|\gtrsim q.$$

The exponent $(d+1)/2$ is known to be sharp in odd dimensions $d\geq 3$, while this conjectured improvement to $d/2$ remains open for all even dimensions.