and hence $$\mu_G(A' +_{\delta+{\varepsilon}} B) \leq \mu_G(A') + \mu_G(B) + o(1).$$ From this we easily verify that $(A',B)$ is a critical pair as claimed.

We can now simplify the problem by observing that if one element $(A,B)$ of a critical pair is already asymptotically equivalent to a Bohr set, then so is the other:

\[p1\] Let $(A,B)$ be a critical pair, and suppose that $B$ is asymptotically equivalent to a Bohr set $\phi^{-1}(J)$. Then $A$ is asymptotically equivalent to a parallel Bohr set $\phi^{-1}(I)$.

By Lemma \[crit-equiv\], we may assume without loss of generality that $B = \phi^{-1}(J)$; also, by translation invariance we may assume that $J = [0,t] \text{ mod } {\mathbb{Z}}$ for some $t$ with $$\label{muat}
\mu_G(A), t, 1 - \mu_G(A) - t \gg 1.$$ As $(A,B)$ is a critical pair, there exists an infinitesimal $\delta>0$ such that the set $C \coloneqq A +_\delta B$ has measure $$\mu_G(C) = \mu_G(A) + \mu_G(B) + o(1) = \mu_G(A) + t + o(1).$$

The set $B$ is invariant with respect to translations in the kernel of $\phi$, so $C$ is similarly invariant, thus $C = \phi^{-1}(E)$ for some measurable subset $E$ of ${\mathbb{R}}/{\mathbb{Z}}$ with $$\label{met}
\mu_{{\mathbb{R}}/{\mathbb{Z}}}(E) = \mu_G(C) = \mu_G(A) + 