An extension of the direction problem

TL;DR

This paper extends the concept of directions determined by point sets in affine spaces to higher-dimensional subspaces, classifies extremal cases, and identifies point sets that do not determine all such subspaces.

Contribution

It introduces a new definition for directions determined by point sets in affine spaces and classifies extremal point sets that fail to determine all subspaces.

Findings

Classified point sets not determining every $k$-subspace in certain cases

Extended the direction problem to higher-dimensional subspaces

Analyzed extremal case where |U|=q^{n-1}

Abstract

Let $U$ be a point set in the $n$-dimensional affine space ${\rm AG}(n,q)$ over the finite field of $q$ elements and $0\leq k\leq n-2$. In this paper we extend the definition of directions determined by $U$: a $k$-dimensional subspace $S_k$ at infinity is determined by $U$ if there is an affine $(k+1)$-dimensional subspace $T_{k+1}$ through $S_k$ such that $U\cap T_{k+1}$ spans $T_{k+1}$. We examine the extremal case $|U|=q^{n-1}$, and classify point sets NOT determining every $k$-subspace in certain cases.