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 $\mathrm{\mid }A\mathrm{\mid }\ge \alpha p$ for some $\alpha \in (0,1)$, we determine the optimal constant $f(\alpha )$ in the inequality $$\max (\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }A\cdot A\mathrm{\mid })\ge (f(\alpha )-o(1))p\mathrm{.}$$ The proof relies on a structural result for sumsets of dense subsets, established via a regularity lemma in general finite abelian groups.