${\varepsilon}$. Then, for any compact subsets $A,B$ of a compact connected abelian group $G = (G,+)$ with $$\mu_G(A), \mu_G(B), 1 - \mu_G(A) - \mu_G(B) \geq {\varepsilon}$$ and $$\mu_G(A+B) \leq \mu_G(A) + \mu_G(B) + \delta,$$ there exist parallel Bohr sets $\phi^{-1}(I), \phi^{-1}(J)$ such that $$\mu_G( A \Delta \phi^{-1}(I) ), \mu_G( B \Delta \phi^{-1}(J) ) \leq {\varepsilon},$$ where $A \Delta B$ denotes the symmetric difference of $A$ and $B$.

In the case $G = {\mathbb{R}}/{\mathbb{Z}}$, this result was recently obtained in [@candela Theorem 1.5] (with a quite sharp dependence between ${\varepsilon}$ and $\delta$), by a different method; see also the earlier work [@mfy], [@fjm]. In the case of a torus $G = ({\mathbb{R}}/{\mathbb{Z}})^d$, when the measures of $A$ and $B$ are small and comparable to each other, this theorem was obtained (again with a sharp dependence between ${\varepsilon}$ and $\delta$) in [@bilu Theorem 1.4].

As a consequence of the above theorem, we can reprove a theorem of Kneser [@kneser Satz 2] classifying when equality holds in :

\[cor\] Let let $A,B$ be non-empty compact subsets of a compact connected abelian group $G$ such that equality holds in . Then at least one of the following statements hold:

-   $\mu_G(A)=0$ or $\mu_G(B) = 