On the discretized sum-product problem

Abstract

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 $|A{|}^{1-\frac{1}{68}}$

13 pages, 0 figures. v3: final version, to appear in IMRN

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