Discretized sum-product for large sets

Abstract

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

arXiv comment12 pages, to appear in Moscow Journal of Combinatorics and Number Theory