An inverse theorem for an inequality of Kneser

Let $G = (G,+)$ be a compact connected abelian group, and let $\mu_G$ denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath and Raikov) establishes the bound $$\mu_G(A + B) \geq \min( \mu_G(A)+\mu_G(B), 1 )$$ whenever $A,B$ are compact subsets of $G$, and $A+B := \{ a+b: a \in A, b \in B \}$ denotes the sumset of $A$ and $B$. Clearly one has equality when $\mu_G(A)+\mu_G(B) \geq 1$. Another way in which equality can be obtained is when $A = \phi^{-1}(I), B = \phi^{-1}(J)$ for some continuous surjective homomorphism $\phi: G \to {\bf R}/{\bf Z}$ and compact arcs $I,J \subset {\bf R}/{\bf Z}$. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then $A,B$ are close to one of the above examples. We also give a more "robust" form of this theorem in which the sumset $A+B$ is replaced by the partial sumset $A +_\varepsilon B :=\{ 1_A * 1_B \geq \varepsilon \}$ for some small $\varepsilon >0$. In a subsequent paper with Joni Ter\"av\"ainen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.