  Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of
whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian
groups, we show that it is no in general.
  On the other hand, we show that every $n$-vertex Cayley graph (and more
generally, vertex-transitive graph) has an orthonormal basis whose coordinates
are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible.
  Our investigation is motivated by a question of Assaf Naor, who proved that
random abelian Cayley graphs are small-set expanders, extending a classic
result of Alon--Roichman. His proof relies on the existence of a bounded
eigenbasis for abelian Cayley graphs, which we now know cannot hold for general
groups. On the other hand, we navigate around this obstruction and extend
Naor's result to nonabelian groups.
