Point Distribution and Perfect Directions in F2 P

Let \( p \geq 3 \) be a prime, \( S \subseteq \mathbb{F}_p^2 \) a nonempty set, and \( w: \mathbb{F}_p^2 \to \mathbb{R} \) a function with \( \operatorname{supp} w = S \). Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most \( \frac{1}{2}|S| \) directions in \( \mathbb{F}_p^2 \) such that for every line \( l \) in any of these directions, one has

\[
\sum_{z \in l} w(z) = \frac{1}{p} \sum_{z \in \mathbb{F}_p^2} w(z),
\]

except if \( S \) itself is a line and \( w \) is constant on \( S \) (in which case all, but one direction have the property in question). The bound \( \frac{1}{2}|S| \) is sharp.

As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.