On the complexity of sets of uniqueness and extended uniqueness

We locate the complexity of the set of closed sets of uniqueness U(G), for G locally compact Lie group and of the set of closed sets of extended uniqueness U_0(G), for G connected abelian Lie group. More concretely, we prove that with respect to the Effros-Borel space, these sets are coanalytic complete. This extends previous results obtained for the case G=T.