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

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 $\rho : G \rightarrow \mbox{U}_d(\mathbf{C})$ 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 \[ \sup_{g \in G} |\langle \rho(g) v, w \rangle| \leq (1 + c \log d)^{-1/2}.\] 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 $\pi/2 - o_{d \rightarrow \infty}(1)$.