# Point distribution and perfect directions in $F_p^2$

## TL;DR

This paper investigates the distribution of points and directions in finite fields, establishing sharp bounds on directions with uniform sums over lines, and applies these results to finite affine plane geometry.

## Contribution

It introduces a new bound on the number of directions with uniform line sums in finite fields, extending previous uncertainty inequalities and providing a new proof for a classical geometric result.

## Key findings

- Maximum of half the set size for directions with uniform sums
- Bound is sharp, with equality cases characterized
- Application to finite affine plane geometry

## 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.