Discretized sum-product for large sets

Abstract

Let $A\subset \left[1,2\right]$ be a $\left(\delta ,\sigma \right)$-set with measure $\left|A\right|={\delta }^{1-\sigma }$ in the sense of Katz and Tao. For $\sigma \in \left(\frac{1}{2},1\right)$ we show that

$$|A+A|+\left|AA\right|\succsim {\delta }^{-c}\left|A\right|$$

for $c=\left(1-\sigma \right)\left(2\sigma -1\right)∕\left(6\sigma +4\right)$. This improves the bound of Guth, Katz, and Zahl for large $\sigma$.