of Mathematics, UCLA\
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author:
- Terence Tao
title: An inverse theorem for an inequality of Kneser
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Introduction
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Throughout this paper, we use $\mu_G$ to denote the Haar probability measure on any compact abelian group $G = (G,+)$; thus for instance $\mu_{{\mathbb{R}}/{\mathbb{Z}}}$ is Lebesgue measure on the unit circle ${\mathbb{R}}/{\mathbb{Z}}$. In [@kneser], Kneser established[^1] the inequality $$\label{kemp}
 \mu_G(A + B) \geq \min( \mu_G(A)+\mu_G(B), 1 )$$ whenever $A,B$ are non-empty compact subsets of a compact connected abelian group $G$, and $A+B \coloneqq \{ a+b: a \in A, b \in B \}$ denotes the sumset of $A$ and $B$. A subsequent result of Kemperman [@kemperman] extended this inequality to compact connected nonabelian groups also, but we restrict attention here to the abelian case. Prior to Macbeath’s result, the case of a circle $G = {\mathbb{R}}/{\mathbb{Z}}$ was obtained by Raikov [@raikov] (and can also be derived by a limiting argument from the Cauchy-Davenport inequality), the case of a torus $G = ({\mathbb{R}}/{\mathbb{Z}})^d$ was obtained by Macbeath [@macbeath], and the case of second countable connected compact groups by Shields [@shields]. The fact that $G$ is connected is crucial, since otherwise $G$ could contain open subgroups 