  Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms
of the characteristic $p$ of $F$ if $p>0$.
  We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg
|A|^6\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due
to Roche-Newton, Rudnev and Shkredov, to
  $$|AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)}\,,\;\;\;\;\mbox{hence} \;\;
\;\;|AA|+|A\pm A|\gg |A|^{1+\frac{2}{9}-o(1)},$$ as well as $$
|AA|^{36}|A-A|^{24} \gg |A|^{73-o(1)}. $$ The latter inequality is
"threshold-breaking", for it shows for $\epsilon>0$, one has $$|AA| \le
|A|^{1+\epsilon}\;\;\;\Rightarrow\;\;\; |A-A|\gg
|A|^{\frac{3}{2}+c(\epsilon)},$$ with $c(\epsilon)>0$ if $\epsilon$ is
sufficiently small.
  This implies that regardless of $\epsilon$, $$|AA-AA|\gg
|A|^{\frac{3}{2}+\frac{1}{56}-o(1)}\,.$$
