Sums and products in finite fields: an integral geometric viewpoint

Derrick Hart, Alex Iosevich

We prove that if $A\subset {\mathbb{F}}_q$ is such that $$|A|>q^{1/2+\frac{1}{2d}},$$ then $${\mathbb{F}}_q^{\ast }\subset d{A}^2=A^2+...+A^{2}d\text{times},$$ where $$A^2=\{a \cdot a': a,a'…