Discrete circle actions: A note using non-standard analysis

In this note we prove that the existence of effective uniformly Lipschitz ℚ/ℤ actions on manifolds (and other spaces) follows from the existence of such Z_n actions. The method of approach is non-standard analysis with all non-trivial transformation group theoretical information concentrated in Newman's theorem; this results in a completely elementary argument. We give examples showing that, in contrast, there are spaces with no S¹ effective actions despite Z_n and hence ℚ/ℤ effective actions.