The sum-product phenomenon for dense subsets of finite fields

TL;DR

This paper establishes the optimal constant in the sum-product inequality for dense subsets of finite fields, using a structural regularity lemma in finite abelian groups.

Contribution

It determines the exact constant in the sum-product inequality for dense subsets of finite fields, extending previous bounds.

Findings

Identifies the optimal constant f(α) in the sum-product inequality for dense subsets.

Uses a structural regularity lemma to analyze sumsets in finite abelian groups.

Provides a precise threshold for sumset and product set sizes in dense regimes.

Abstract

Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq \alpha p$ for some $\alpha \in (0,1)$, we determine the optimal constant $f(\alpha)$ in the inequality $$ \max(|A+A|, |A\cdot A|) \geq (f(\alpha) - o(1))p. $$ The proof relies on a structural result for sumsets of dense subsets, established via a regularity lemma in general finite abelian groups.