Distinct spreads in vector spaces over finite fields

In this short note, we study the distribution of spreads in a point set P⊆Fqd, which are analogous to angles in Euclidean space. More precisely, we prove that, for any ε>0, if |P|≥(1+ε)q⌈d∕2⌉, then P determines a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets P⊂Fqd of size |P|=q⌈d∕2⌉ that determine at most one spread.