Erdős–Falconer distance conjecture in even-dimensional finite vector spaces

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $d\geq 2$ be even, and equip $\mathbb{F}_q^d$ with the quadratic form

$$\|\mathbf{x}\|:=x_1^2+\cdots+x_d^2.$$

For $E\subseteq\mathbb{F}_q^d$, define its distance set by

$$\Delta(E):=\{\|\mathbf{x}-\mathbf{y}\|:\mathbf{x},\mathbf{y}\in E\}\subseteq\mathbb{F}_q.$$

Let $\alpha(d)$ be the smallest exponent such that whenever $E\subseteq\mathbb{F}_q^d$ satisfies $|E|\gg q^{\alpha(d)}$, one has $|\Delta(E)|\gg q$. Erdős–Falconer distance conjecture. If $d\geq 2$ is even, then

$$\alpha(d)=\frac{d}{2}.$$

This is the finite-field analogue of the Erdős–Falconer distance conjecture from geometric measure theory. The conjecture specifies the expected sharp threshold for positive-proportion distance sets in even dimensions.