  We investigate to what extent a minimal topological dynamical system is
uniquely determined by a set of return times to some open set. We show that in
many situations this is indeed the case as long as the closure of this open set
has no non-trivial translational symmetries. For instance, we show that under
this assumption two Kronecker systems with the same set of return times must be
isomorphic. More generally, we show that if a minimal dynamical system has a
set of return times that coincides with a set of return times to some open set
in a Kronecker system with translationarily asymmetric closure, then that
Kronecker system must be a factor. We also study similar problems involving
Nilsystems and polynomial return times. We state a number of questions on
whether these results extend to other homogeneous spaces and transitive group
actions, some of which are already interesting for finite groups.
