On the discretized sum-product problem

Larry Guth, Nets Hawk Katz, Joshua Zahl

Published 2018-04-06, updated 2019-06-19Version 2

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(\delta,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$