$\mathbb F$ is finite field, $\mathbb A$ is a subset of $\mathbb F$, and $|\mathbb A|>|\mathbb F|^{\frac 34} $. Proof $\forall x \in \mathbb F$, there exist $a,b,c,d,e,f\in\mathbb A$ that makes $x = ab+cd+ef$.
A possible route may be to use Kneser's theorem to get a handle on the available number of products $ab$, and then apply Cauchy-Davenport or Kneser again on the additive side. Can this be made to work, or is a different approach required?
This is a lemma in the paper Mordell's exponential sum estimate revisited by Jean Bourgain. The author proves it using Fourier analytic methods applied to finite abelian groups.