Special directions on the finite affine plane

TL;DR

This paper investigates the properties of sets in finite affine planes over prime characteristic fields, focusing on the number of special directions, and provides classifications and construction methods for sets with specific numbers of directions.

Contribution

It proves the non-existence of sets with exactly two special directions, characterizes sets with three directions, and introduces methods to construct minimal sets with four directions for small primes.

Findings

No sets with exactly two special directions exist.

Sets with exactly three special directions are characterized.

Methods for constructing minimal sets with four directions are developed.

Abstract

In this paper we study the number of special directions of sets of cardinality divisible by $p$ on a finite plane of characteristic $p$, where $p$ is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly 3 special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that has exactly 4 special directions for small values of $p$.