Discretized sum-product for large sets

Changhao Chen

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