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 $\bT$ or $\bT \rtimes \{-1,1\}$. This extends a classical result of Kneser.

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