0$.

-   $A,B$ are parallel Bohr sets.

-   $\mu_G(A) + \mu_G(B) \geq 1$.

We prove this corollary in Section \[cor-sec\].

Much as can be deduced from Corollary \[kemp-cor\], Theorem \[inv-1\] will be deduced from the following variant:

\[inv-2\] Let ${\varepsilon}>0$, and suppose that $\delta>0$ is sufficiently small depending on ${\varepsilon}$. Then, for any measurable subsets $A,B$ of a compact connected abelian group $G$ with $$\mu_G(A), \mu_G(B), 1 - \mu_G(A) - \mu_G(B) \geq {\varepsilon}$$ and $$\mu_G(A +_\delta B) \leq \mu_G(A) + \mu_G(B) + \delta,$$ there exist parallel Bohr sets $\phi^{-1}(I), \phi^{-1}(J)$ such that $$\mu_G( A \Delta \phi^{-1}(I) ), \mu_G( B \Delta \phi^{-1}(J) ) \leq {\varepsilon}.$$

Since $A +_\delta B$ is clearly contained in $A+B$, it is immediate that Theorem \[inv-2\] implies Theorem \[inv-1\].

The proof of Theorem \[inv-2\] can be outlined as follows. To simplify this outline, let us ignore all the ${\varepsilon}$ and $\delta$ errors, in particular pretending that the partial sumset $A +_\delta B$ is the same as the full sumset $A+B$. Let us informally call a pair $(A,B)$ a “critical pair” if the conditions of Theorem \[inv-2\] are obeyed. By using “submodularity inequalities” such as $$\mu_G( (A_1 \cup A_2) + B) + \mu_G( (A_1 \cap A_2) + B) 