Stronger sum-product inequalities for small sets

TL;DR

This paper improves sum-product inequalities in finite fields, establishing stronger bounds that demonstrate small product sets imply large difference sets, advancing understanding of additive and multiplicative structure in finite fields.

Contribution

The paper introduces significantly strengthened sum-product inequalities for small sets in finite fields, surpassing previous thresholds and revealing new structural implications.

Findings

Established new threshold-breaking sum-product inequalities

Proved that small product sets imply large difference sets

Demonstrated bounds that improve understanding of finite field set structures

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