  We show that for every $1 \le k \le d/(\log d)^C$, every finite transitive
set of unit vectors in $\mathbb{R}^d$ lies within distance $O(1/\sqrt{\log
(d/k)})$ of some codimension $k$ subspace, and this distance bound is best
possible. This extends a result of Ben Green, who proved it for $k=1$.
