# Subsets of $\mathbb{F}^*_p$ with only small products or ratios

## TL;DR

This paper investigates the structure of subsets of the finite field multiplicative group with restrictions on their products or ratios, providing estimates for the number of elements satisfying certain modular product conditions.

## Contribution

It introduces new bounds and estimates for the number of elements in subsets of finite fields with small product or ratio properties, extending previous results.

## Key findings

- Derived bounds for subset sizes with small product conditions
- Extended analysis to ratio-based subset properties
- Generalized results to broader classes of finite field subsets

## Abstract

Let $p$ be a fixed prime. We estimate the number of elements of a set $A \subseteq \mathbb{F}^*_p$ for which $$ s_1s_2 \equiv a \pmod{p} \quad \mbox{for some}\quad a \in [-X,X] \quad \mbox{for all}\quad s_1,s_2 \in A. $$ We also consider variations and generalizations.