New sum product type estimates

TL;DR

This paper extends sum-product estimates to complex numbers, providing new lower bounds for sum, difference, product, and ratio sets, generalizing real results and improving known bounds using combinatorial geometry techniques.

Contribution

It introduces new sum-product type bounds for complex sets, generalizing Solymosi's real-plane approach and enhancing bounds involving difference sets through advanced combinatorial methods.

Findings

Sum set bounds match Solymosi's real results up to constants.

Difference set bounds are slightly weaker but improved over previous estimates.

Combines Szemerédi-Trotter theorem with arithmetic combinatorics techniques.

Abstract

New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising his approach to the complex plane. The bounds involving the difference set are slightly weaker. They improve on the best known ones, including the case $A\subset \R$, which also due to Solymosi, by means of combining the use of the Szemer\'edi-Trotter theorem with an arithmetic combinatorics technique.