Distinct spreads in vector spaces over finite fields

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.