# Discretized sum-product for large sets

## TL;DR

This paper proves a new lower bound on the sum and product sets of certain discretized sets in [1, 2], improving previous results for larger values of .

## Contribution

It introduces an improved bound on sum and product sets for -sets, advancing the understanding of sum-product phenomena in discretized settings.

## Key findings

- Established a lower bound |A+A|+|AA|  ^{-c}|A| for   in (1/2, 1)
- Improved upon previous bounds by Guth, Katz, and Zahl for large 
- Demonstrated the effectiveness of discretized sum-product estimates in additive combinatorics.

## Abstract

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for $c=\frac{(1-\sigma)(2\sigma-1)}{6\sigma+4}$. This improves the bound of Guth, Katz, and Zahl for large $\sigma$.