On products of shifts in arbitrary fields

Abstract

We adapt the approach of Rudnev, Shakan, and Shkredov (2018) to prove that in an arbitrary field $\mathbb{F}$, for all $A\subset \mathbb{F}$ finite with $\left|A\right|<p^{1∕4}$ if $p:=Char\left(\mathbb{F}\right)$ is positive, we have

$$\left|A\left(A+1\right)\right|\gg \frac{|A{|}^{11∕9}}{{\left(log\left|A\right|\right)}^{7∕6}},\left|AA\right|+\left|\left(A+1\right)\left(A+1\right)\right|\gg \frac{|A{|}^{11∕9}}{{\left(log\left|A\right|\right)}^{7∕6}}.$$

This improves upon the exponent of $\frac{6}{5}$ given by an incidence theorem of Stevens and de Zeeuw.