# Some multiplicative equations in finite fields

## TL;DR

This paper provides new bounds on the number of solutions to multiplicative equations in finite fields, especially for structured sets like arithmetic progressions and boxes, extending previous methods.

## Contribution

It introduces sharp bounds for multiplicative energy in generalized arithmetic progressions and boxes, extending prior techniques to broader settings.

## Key findings

- Sharp bounds for multiplicative energy in prime fields
- Extension of methods to arbitrary finite fields
- Improved estimates for structured sets

## Abstract

In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating the multiplicative energy of generalized arithmetic progressions in prime fields and of boxes in arbitrary finite fields and obtain sharp bounds in more general scenarios than previously known. Our arguments extend some ideas of Konyagin and Bourgain and Chang into new settings.