The cylindrical width of transitive sets

Abstract

We show that for every 1 ≤ k ≤ d/(log d)C, for some absolute constant C, that every finite transitive set of unit vectors in ℝd lies within distance \(O\left({1/\sqrt {\log \left({d/k} \right)}} \right)\) 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.