Thresholds in the Lattice of Subspaces of $({\mathbb{F}}_q{)}^n$

Let $Q$ be an ideal (downward-closed set) in the lattice of linear subspaces of $({\mathbb{F}}_q{)}^n$, ordered by inclusion. For $0\le k\le n$, let ${\mu }_k(Q)$ denote the fraction of $k$-dimensional subspaces that belong to $Q$. We show that these densities satisfy $${\mu }_k(Q)=\frac{1}{1+z}⟹{\mu }_{k+1}(Q)\le \frac{1}{1+qz}\mathrm{.}$$ This implies a sharp threshold theorem: if ${\mu }_k(Q)\le 1-\epsilon$, then $\mu_\ell(Q) \le \varepsilon$for$\ell = k + O(\log_q(1/\varepsilon))$.