Stronger sum-product inequalities for small sets

Abstract: 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 $\epsilon$, $$|AA-AA|\gg |A|^{\frac{3}{2}+\frac{1}{56}-o(1)}\,.$$