The sum-product phenomenon for dense subsets of finite fields

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.