On the width of transitive sets: bounds on matrix coefficients of finite groups

Abstract

We say that a finite subset of the unit sphere in ${\mathbf{R}}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(\log d{)}^{-1/2}$. This is a consequence of the following result: If $G$ is a finite group and $ρ: G \rightarrow \mbox{U}_d(\mathbf{C})$ a unitary representation, and if $v\in {\mathbf{C}}^d$ is a unit vector, there is another unit vector $w\in {\mathbf{C}}^d$ such that \[ \sup_{g \in G} |\langle ρ(g) v, w \rangle| \leq (1 + c \log d)^{-1/2}.\] These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient $S({\mathbf{R}}^d)/G$ of the unit sphere by a finite group $G$ of isometries is at least $\pi /2-o_{d\to \infty }(1)$.

35 pages, corrected two significant errors drawn to my attention by Ashwin Sah, Mehtaab Sawhney and Yufei Zhao