Let
$A\subset \left[1,2\right]$ be a
$\left(\delta ,\sigma \right)$set with measure
$\leftA\right={\delta}^{1\sigma}$ in the sense of
Katz and Tao. For
$\sigma \in \left(\frac{1}{2},1\right)$
we show that
$$A+A+\leftAA\right\succsim {\delta}^{c}\leftA\right$$
for
$c=\left(1\sigma \right)\left(2\sigma 1\right)\u2215\left(6\sigma +4\right)$.
This improves the bound of Guth, Katz, and Zahl for large
$\sigma $.
