The cylindrical width of transitive sets

Abstract

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 $k=1$.

17 pages