On the Discretized Sum-Product Problem

Abstract

<jats:title>Abstract</jats:title> <jats:p>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 $|A|^{1-\frac {1}{68}}$.</jats:p>