On the discretized sum-product problem

TL;DR

This paper presents a new proof of the discretized ring theorem, demonstrating that for certain real sets, either their sum or product set must have significantly large measure, advancing understanding in additive combinatorics.

Contribution

It provides a novel proof of the discretized ring theorem, improving bounds on the measure of sum or product sets for specific real sets.

Findings

Either the sum set or the product set of a $( ext{delta},1/2)_1$-set has measure at least $|A|^{1-rac{1}{68}}$

The new proof simplifies previous approaches to the discretized ring theorem

Enhances bounds in additive combinatorics for real sets

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