Distinct spreads in vector spaces over finite fields

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 $\epsilon >0$, if $|\mathcal{P}|\ge (1+\epsilon )q^{\lceil d/2\rceil }$, 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.