Compact groups with high commuting probability of monothetic subgroups

If $H$ is a subgroup of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $G$ is denoted by $Pr(H,G)$. Let $⟨g⟩$ stand for the monothetic subgroup generated by an element $g\in G$ and let $K$ be a subgroup of $G$. We prove that $Pr(⟨x⟩,G)>0$ for any $x\in K$ if and only if $G$ has an open normal subgroup $T$ such that $K\mathrm{/}{C}_K(T)$ is torsion. In particular, $Pr(⟨x⟩,G)>0$ for any $x\in G$ if and only if $G$ is virtually central-by-torsion, that is, there is an open normal subgroup $T$ such that $G\mathrm{/}Z(T)$ is torsion. We also deduce a number of corollaries of this result.