Point distribution and perfect directions in $F_p^2$

Abstract

Let $p\ge 3$ be a prime, $S\subseteq\mathbb F_p^2$ a nonempty set, and $w\colon\mathbb F_p^2\to\mathbb R$ a function with $\mathrm{supp}\, w=S$. Applying an uncertainty inequality due to Andr\'as Bir\'o and the present author, we show that there are at most $\frac12|S|$ directions in $\mathbb F_p^2$ such that for every line $l$ in any of these directions, one has $$ \sum_{z\in l} w(z) = \frac1p\sum_{z\in\mathbb F_p^2} w(z), $$ 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 $\frac12|S|$ 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.