On the complexity of sets of uniqueness and extended uniqueness

TL;DR

This paper investigates the descriptive set-theoretic complexity of the sets of uniqueness and extended uniqueness in locally compact Lie groups, establishing their coanalytic completeness in the Effros-Borel space.

Contribution

It extends prior results by proving the coanalytic completeness of these sets for a broader class of Lie groups beyond the circle group.

Findings

U(G) and U_0(G) are coanalytic complete in the Effros-Borel space

Extends previous results from G=T to more general Lie groups

Provides a descriptive set-theoretic classification of these sets

Abstract

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.