On the discretized sum-product problem

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 $(\delta ,1\mathrm{/}2{)}_1$-set in the sense of Katz and Tao, then either $A+A$ or $A\mathrm{.}A$ must have measure at least $\mathrm{\mid }A{\mathrm{\mid }}^{1-\frac{1}{68}}$