I'm looking for references about the following aspect of cyclic vectors for regular representations.
Let $K$ be a compact Lie group. Let $K$ act on $L^2(K)$ by the left regular representation. Then $L^2(K)$ has a cyclic vector $v$, so that $K\cdot v$ has dense linear span in $L^2(K)$ (for example one can find a proof of this in the 1971 paper of Greenleaf and Moskowitz "Cyclic Vectors for Representations of Locally Compact Groups").
Question: do we have any control over linear dependence of the set $A=\{kv:k\in K\}$? More precisely, is it possible to choose the cyclic vector $v$ so that $A$ is linearly independent; if not, is it possible to choose $v$ so that there exists some measure-zero set $S\subseteq K$ so that $A'=\{kv:k\in K\backslash S\}$ is linearly independent?
PS: I noticed that the paper "Linear Dependency of Translations and Square Integrable Representations" by Linnell, Puls and Roman discusses the problem of finding $v$ so that $A$ becomes linearly dependent; I'm interested in the case when $v$ is cyclic.