If $A \subseteq \mathbb{Z}/p\mathbb{Z}$, and $|A| > \frac{p}{3}$, then are there any nontrivial lower bounds for $|AA|$?
Where $AA=\{a_{1} \cdot a_2:a_1,a_2 \in A\}$, and $p$ is prime.
Writing out manually some multiplication results for some low $p$ and big enough $A$ gives the impression that if $|A| > \frac{p}{3}$, or so, then it should be the case that $|AA| \geq p-1$.
It seems to me that I'm missing some basic knowledge about the multiplication in the $\mathbb{Z}/p\mathbb{Z}$, since there are clear patterns in the respective multiplication table for $AA$, as above, like the fact that it is symmetric over both diagonals, or that each row and column consist of the different numbers, etc.