On rich and poor directions determined by a subset of a finite plane

We generalize to sets with cardinality more than $p$ a theorem of R'edei and Sz\H{o}nyi on the number of directions determined by a subset $U$ of the finite plane ${\mathbb{F}}_p^2$. A $U$-rich line is a line that meets $U$ in at least $\mathrm{\#}U\mathrm{/}p+1$ points, while a $U$-poor line is one that meets $U$ in at most $\mathrm{\#}U\mathrm{/}p-1$ points. The slopes of the $U$-rich and $U$-poor lines are called $U$-special directions. We show that either $U$ is contained in the union of $n=\lceil \mathrm{\#}U\mathrm{/}p\rceil$ lines, or it determines many' $U$-special directions. The core of our proof is a version of the polynomial method in which we study iterated partial derivatives of the R\'edei polynomial to take into account the multiplicity’ of the directions determined by $U$.