The "gaps" in $\{ \{ n \alpha\}: 0 < n \leq N \}$ take up to three values while random numbers on the circle have Poisson distributed gaps. The only example of a Nilmanifold I'll mention is the Heisenberg nilmanifold . $$ \left( \begin{array}{ccc} 1 & \mathbb{R} & \mathbb{R} \\ 0