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arXiv · 2604.17117

The sum-product phenomenon for dense subsets of finite fields

Xuancheng Shao
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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.

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Xuancheng Shao. 2026-04-18. The sum-product phenomenon for dense subsets of finite fields. https://arxiv.org/abs/2604.17117

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