  There exists an absolute constant $\delta > 0$ such that for all $q$ and all
subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if
$|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in
A\}| > \frac{q}{2}. \] Any $\delta < 1/13,542$ suffices for sufficiently large
$q$. This improves the condition $|A| > q^{2/3}$, due to Bennett, Hart,
Iosevich, Pakianathan, and Rudnev, that is typical for such questions.
  Our proof is based on a qualitatively optimal characterisation of sets $A,X
\subseteq \mathbb{F}_q$ for which the number of solutions to the equation \[
(a_1-a_2) = x (a_3-a_4) \, , \; a_1,a_2, a_3, a_4 \in A, x \in X \] is nearly
maximum.
  A key ingredient is determining exact algebraic structure of sets $A, X$ for
which $|A + XA|$ is nearly minimum, which refines a result of Bourgain and
Glibichuk using work of Gill, Helfgott, and Tao.
  We also prove a stronger statement for \[ (A-B)(C-D) = \{ (a -b) (c-d) : a
\in A, b \in B, c \in C, d \in D\} \] when $A,B,C,D$ are sets in a prime field,
generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.
