In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself.

Let ${\mathbb F}$ be a finite field, and suppose that $\varphi_0\colon{\mathbb F}\mapsto{\mathbb F}$ is a function from ${\mathbb F}$ to itself. For each $a\in{\mathbb F}$, consider the function $\varphi_a\colon{\mathbb F}\mapsto{\mathbb F}$ defined by $\varphi_a(x)=\varphi_0(x)+ax$ ($x\in{\mathbb F}$).

Is it true that if $q:=|{\mathbb F}|$ is even, then there exists $a\in{\mathbb F}$ such that the image of $\varphi_a$ has size larger than $2q/3$?

(A negative answer would yield an improvement on the known bounds for the smallest size of a Kakeya set in the vector spaces ${\mathbb F}^n$.)

It may be worth explaining where the coefficient $2/3$ comes from. In the aforementioned paper, we show that if $q$ is an even power of $2$, then for $\varphi_0(x)=x^3$ one has $\max_a |\varphi_a({\mathbb F})|\le(2q+1)/3$, whereas if $q$ is an odd power of $2$, then for $\varphi_0(x)=x^{q-2}+x^2$ one has $\max_a |\varphi_a({\mathbb F})|\le2(q+\sqrt q+1)/3$. The question is whether one can get better bounds for an appropriate choice of the function $\varphi_0$.