# Some remarks on the asymmetric sum--product phenomenon

## TL;DR

This paper investigates the asymmetric sum-product phenomenon in finite fields, providing new quantitative bounds on the maximum of certain sum and product sets when the involved subsets differ greatly in size.

## Contribution

It introduces novel bounds based on higher energy observations, advancing understanding of sum-product behavior in asymmetric finite set regimes.

## Key findings

- Established lower bounds on max{|AB|, |A+C|} and similar expressions.
- Demonstrated bounds hold when subset sizes differ significantly.
- Utilized higher energy techniques to derive quantitative results.

## Abstract

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $\max\{|AB|, |A+C| \}$ and $\max\{|(A+\alpha)B|, |A+C|\}$, $\alpha \neq 0$ in the regime when the sizes of finite subsets $A,B,C$ of a field differ significantly.