Stronger sum-product inequalities for small sets

Abstract:

Let $F$ be a field and let 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 \begin{equation*} |AA|^3 |A\pm A|^2 \gg |A|^6 ,\;\;\;\;\text {hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac {1}{5}}, \end{equation*} due to Roche–Newton, Rudnev, and Shkredov, to \begin{equation*} |AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)} ,\;\;\;\;\text {hence} \;\; \;\;|AA|+|A\pm A|\gg |A|^{1+\frac {2}{9}-o(1)}, \end{equation*} as well as \begin{equation*} |AA|^{36}|A-A|^{24} \gg |A|^{73-o(1)}. \end{equation*} The latter inequality is “threshold–breaking”, for it shows for $\epsilon >0$, one has \begin{equation*} |AA| \leq |A|^{1+\epsilon }\;\;\;\Rightarrow \;\;\; |A-A|\gg |A|^{\frac {3}{2}+c(\epsilon )}, \end{equation*} with $c(\epsilon )>0$ if $\epsilon$ is sufficiently small.