On sum sets of sets, having small product set

TL;DR

This paper improves bounds on the sum-product problem in real numbers, showing that either the sum set or the product set of a set A must be significantly large, especially for sets with small product sets.

Contribution

It provides a stronger lower bound on the maximum of sum and product sets, advancing the understanding of sum-product phenomena for sets with small product sets.

Findings

Established that max{|A+A|,|AA|} \\gg |A|^{4/3+c} for some c>0.

Improved previous bounds for sets with small product sets in R.

Provided new lower bounds for sums of sets with small product sets.

Abstract

We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets A from R with |AA| \le |A|^{4/3}.