Iosevich–Rudnev distance-set conjecture over finite fields

Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$, where $q$ is odd. For $E\subseteq\mathbb F_q^d$, define the distance set by

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

where

$$||\alpha||=\sum_{j=1}^d\alpha_j^2$$

for $\alpha=(\alpha_1,\ldots,\alpha_d)\in\mathbb F_q^d$, and $|\Delta(E)|\sim q$ means that the distance set has size comparable to $q$. Iosevich–Rudnev's conjecture. Let $E\subseteq\mathbb F_q^d$. Suppose that $d\geq 2$ is even or $d,q\equiv 3\pmod{4}$. Then, if $|E|\geq Cq^{d/2}$ for a sufficiently large constant $C>0$, one has

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

The previously known threshold is $|E|\geq Cq^{(d+1)/2}$; the conjecture predicts the improved exponent $d/2$ in the stated cases. The claim concerns when a set determines a positive proportion of all possible finite-field distances and remains unresolved in the source.