Even-dimensional two-set Erdős–Falconer distance conjecture

Let $d\ge 2$ be an even integer, let $A,B\subseteq\mathbb F_q^d$, and define

$$\Delta(A,B)=\{\|x-y\|:x\in A,\ y\in B\},$$

where $\|v\|=\sum_{i=1}^dv_i^2$. Two-set Erdős–Falconer conjecture. If

$$|A||B|\ge Cq^d$$

for a sufficiently large constant $C$ independent of $q$, then

$$|\Delta(A,B)|\sim q.$$

This is the conjectured sharp product-size threshold for the generalized distance problem in higher even dimensions. The paper notes that the exponent $d+1$ is the best known general bound for even dimensions $d\ge 4$, while smaller exponents are known in dimension $2$.