Non-existence of sets with few special directions

Source abstract

Let $p$ be an odd prime and let $S\subseteq\mathbb{F}_p^2$ have cardinality divisible by $p$. We prove that, for all sufficiently large primes $p$, no such set has exactly four special directions, and obtain a conditional extension to larger numbers of special directions under an affine-independence assumption on the corresponding projection functions. A separate second-moment argument shows more generally that, for every fixed $k\ge4$, no subset of $\mathbb{F}_p^2$ has exactly $k$ special directions once $p$ is sufficiently large. In fact, the result holds uniformly for $k$ up to a positive constant times $\sqrt p/\log p$. In contrast, for multisets every prescribed collection of at most $p$ directions can occur as the set of special directions of a $\{0,1,2\}$-valued multiset on $\mathbb{F}_p^2$.