Discretized sum-product for large sets

Let $A\subset [1,2]$ be a $(\delta ,\sigma )$-set with measure $\mathrm{\mid }A\mathrm{\mid }={\delta }^{1-\sigma }$ in the sense of Katz and Tao. For $\sigma \in (1\mathrm{/}2,1)$ we show that $$\mathrm{\mid }A+A\mathrm{\mid }+\mathrm{\mid }AA\mathrm{\mid }⪆{\delta }^{-c}\mathrm{\mid }A\mathrm{\mid },$$ for $c=\frac{(1-\sigma )(2\sigma -1)}{6\sigma +4}$. This improves the bound of Guth, Katz, and Zahl for large $\sigma$.