Distinct spreads in vector spaces over finite fields

TL;DR

This paper investigates the distribution of spreads, analogous to angles, in finite field vector spaces and establishes conditions under which a point set determines a positive proportion of all spreads.

Contribution

It proves that large enough point sets in finite fields generate a positive proportion of spreads, and shows the tightness of these bounds with explicit constructions.

Findings

Sets of size greater than (1+ε)q^{ceil(d/2)} determine many spreads.

Sets of size q^{ceil(d/2)} can determine at most one spread.

Results are tight, with explicit examples matching bounds.

Abstract

In this short note, we study the distribution of spreads in a point set $\mathcal{P} \subseteq \mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $\varepsilon > 0$, if $|\mathcal{P}| \geq (1+\varepsilon) q^{\lceil d/2 \rceil}$, then $\mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $\mathcal{P} \subset \mathbb{F}_q^d$ of size $|\mathcal{P}| = q^{\lceil d/2 \rceil}$ that determine at most one spread.