On the number of directions formed by Cartesian products in ${\mathbb{F}}_{p^2}^2$

We prove a lower bound on the number of directions determined by Cartesian products $A\times A$ in the affine plane over the finite field ${\mathbb{F}}_{p^2}$. Our lower bound holds for sets of size $p^{2\mathrm{/}3}<\mathrm{\mid }A\mathrm{\mid }<p$, which are not contained in any affine copy of ${\mathbb{F}}_p$. The proof combines a structural result of Li and Roche-Newton on the set of directions formed by Cartesian products with a lower bound of Fancsali, Sziklai and Takáts. A key step shows that, unless the set of directions exhibits closure properties forcing subfield structure, one obtains a direction for which an algebraic multiplicity parameter in the latter theorem can be made explicit.