For the dual object of any compact, non-abelian group, the theory of $I_0$ sets (sets of interpolation by Fourier transforms of discrete measures) is quite different from that for abelian compact groups. In general, infinite $I_0$ sets need not exist. However, when the dual object contains an infinite (local) Sidon set, we prove that this set itself has an infinite $I_0$ subset.

The proof is constructive and includes some key examples of $I_0$ sets: certain sets of representations of bounded degree and, for products of simple Lie groups, the set of self-representations of the factor groups and their conjugates (the FTR sets).