Point distribution and perfect directions in $F_p^2$

Let $p\ge 3$ be a prime, $S\subseteq {\mathbb{F}}_p^2$ a nonempty set, and $w\text{\hspace{0.17em}⁣}:{\mathbb{F}}_p^2\to \mathbb{R}$ a function with $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}w=S$. Applying an uncertainty inequality due to Andr'as Bir'o and the present author, we show that there are at most $\frac{1}{2}\mathrm{\mid }S\mathrm{\mid }$ directions in $\mathbb F_p^2$suchthatforeveryline$l$inanyofthesedirections,onehas$${\sum }_{z\in l}w(z)=\frac{1}{p}{\sum }_{z\in {\mathbb{F}}_p^2}w(z),$$exceptif$S$itselfisalineand$w$isconstanton$S$ (in which case all, but one direction have the property in question). The bound $\frac{1}{2}\mathrm{\mid }S\mathrm{\mid }$ is sharp. As an application, we give a new proof of a result of R'edei-Megyesi about the number of directions determined by a set in a finite affine plane.