Small product sets in compact groups

We show in this paper that a sub-critical pair $(A,B)$ of sufficiently “spread-out” Borel sets in a compact and second countable group $K$ with an \emph{abelian} identity component, must reduce to a Sturmian pair in either $\text{bT}$ or $\text{bT}⋊\{-1,1\}$. This extends a classical result of Kneser.