On the structure of the directions not determined by a large affine point set

TL;DR

This paper investigates the structure of undetermined directions in large affine point sets and applies findings to partial ovoids in partial geometries.

Contribution

It provides new structural insights into directions not determined by large affine point sets and links these to partial ovoids in geometrical structures.

Findings

Characterizes the set of undetermined directions in large affine point sets.

Establishes a connection between undetermined directions and partial ovoids.

Offers a structural description applicable to affine geometries.

Abstract

Given a point set $U$ in an $n$-dimensional affine space of size $q^{n-1}-\varepsilon$, we obtain information on the structure of the set of directions that are not determined by $U$, and we describe an application in the theory of partial ovoids of certain partial geometries.