\ll y$, $y \gg x$, or $x = O(y)$. Two measurable subsets $A = A_n$, $B = B_n$ of a compact abelian group $G = G_n$ are said to be *asymptotically equivalent* if one has $\mu_G( A \Delta B ) = o(1)$. This is clearly an equivalence relation.

Theorem \[inv-2\] can now be deduced from the following variant:

\[inv-3\] Let $A = A_n, B = B_n$ be measurable subsets of a sequence $G = G_n$ of compact connected abelian groups with $$\label{muab}
 \mu_G(A), \mu_G(B), 1 - \mu_G(A) - \mu_G(B) \gg 1$$ and $$\label{muab-2}
 \mu_G(A +_\delta B) \leq \mu_G(A) + \mu_G(B) + o(1)$$ for some infinitesimal $\delta > 0$. Then there exist parallel Bohr sets $\phi^{-1}(I) = \phi_n^{-1}(I_n)$ and $\phi^{-1}(J) = \phi_n^{-1}(J_n)$ in $G = G_n$ such that $A$ and $B$ are asymptotically equivalent to $\phi^{-1}(I), \phi^{-1}(J)$ respectively.

Let us assume Theorem \[inv-3\] for now and see how it implies Theorem \[inv-2\] (and hence also Theorem \[inv-1\]). Suppose for contradiction that Theorem \[inv-2\] fails. Carefully negating the quantifiers, and applying the axiom of choice, we conclude that there exists an ${\varepsilon}>0$, such that for every natural number $n$ there are measurable subsets $A = A_n, B = B_n$ of a compact 