In $U(1) = \{z \in \mathbb{C} : |z| = 1\}$, it is well known and easy to see that the set of $z$ so that $ z^n = 1 $ for some $n \in \mathbb{Z}_+$ are dense. Does this fact generalize to the group $U(d) = \{U \in \mathbb{C}^{d \times d} : UU^* = U^*U = I\}$?
What about $ U(\mathcal{H})$, the group of unitary operators on a separable, infinite-dimensional Hilbert space? It seems reasonable to expect that the claim holds for $U(d)$, but not for $ U(\mathcal{H}) $, but I'm not sure.