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
$\leftA\right<{p}^{1\u22154}$
if
$p:=Char\left(\mathbb{F}\right)$ is
positive, we have
$$\leftA\left(A+1\right)\right\gg \frac{A{}^{11\u22159}}{{\left(log\leftA\right\right)}^{7\u22156}},\phantom{\rule{1em}{0ex}}\leftAA\right+\left\left(A+1\right)\left(A+1\right)\right\gg \frac{A{}^{11\u22159}}{{\left(log\leftA\right\right)}^{7\u22156}}.$$
This improves upon the exponent of
$\frac{6}{5}$
given by an incidence theorem of Stevens and de Zeeuw.
