In connection with [this MO post][1], here is a question somewhat
implicitly contained in a [joint paper of S. Kopparty, S. Saraf, M.
Sudan, and myself][2].

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 ${\mathbb F}^n$.)

[1]:
http://mathoverflow.net/questions/102725/a-mixing-property-of-linear-map-over-finite-fields

[2]: http://arxiv.org/pdf/1003.3736.pdf