connected abelian group $G = G_n$ such that for every $n$ one has $$\mu_G(A), \mu_G(B), 1 - \mu_G(A) - \mu_G(B) \geq {\varepsilon}$$ and $$\mu_G(A +_{1/n} B) \leq \mu_G(A) + \mu_G(B) + \frac{1}{n},$$ but such that for each $n$, there do *not* exist parallel Bohr sets $\phi_n^{-1}(I_n), \phi_n^{-1}(J_n)$ such that $$\mu_G( A_n \Delta \phi_n^{-1}(I_n) ), \mu_G( B_n \Delta \phi^{-1}(J_n) ) \leq {\varepsilon}.$$ By applying Theorem \[inv-3\] with the infinitesimal $\delta = \delta_n \coloneqq \frac{1}{n}$, we know that $A,B$ are asymptotically equivalent respectively to parallel Bohr sets $\phi^{-1}(I), \phi^{-1}(J)$. But by taking $n$ large enough, this contradicts the previous statement.

It remains to prove Theorem \[inv-3\]. One of the main reasons of passing to this formulation is that it allows for[^2] the following convenient definition. In the sequel $G = G_n$ is understood to be a sequence of compact connected abelian groups with probability Haar measure $\mu = \mu_n$. A pair $(A,B)$ of measurable subsets of $G$ is said to be a *critical pair*[^3] if one has the properties , for some infinitesimal $\delta>0$. Our goal is thus to prove that every critical pair is equivalent to a pair of parallel Bohr sets.

It turns out that the space of critical pairs 