the sense that $\mu_G(kC) \approx k\mu_G(C)$ for all bounded $k$, which by existing tools in inverse sumset theory (in particular using arguments of Schoen [@schoen] and Green-Ruzsa [@rect], [@green]) can be used to show that $C$ is very close to a Bohr set. As $(A,C)$ is a critical pair, some elementary analysis can then be deployed to show that $A$ is very close to a Bohr set parallel to $C$, and then as $(A,B)$ is also critical, $B$ is also very close to a Bohr set parallel to $A$, giving the claim.

In order to make notions such as “critical pair” rigorous, it will be convenient to use the language of “cheap nonstandard analysis” [@cheap], working with a sequence $(A,B) = (A_n,B_n)$ of pairs in a sequence $G = G_n$ of groups, rather than with a single pair in a single group, so that asymptotic notation such as $o(1)$ can be usefully deployed. It should however be possible to reformulate the arguments below without this language, at the cost of having to pay significantly more attention to various ${\varepsilon}$ and $\delta$ type parameters.

In a subsequent paper with Joni Teräväinen, we will combine this theorem with the structural theory of 