Semicontinuity of structure for small sumsets in compact abelian groups

We study pairs of subsets $A, B$ of a compact abelian group $G$ where the sumset $A+B:=\{a+b: a\in A, b\in B\}$ is small. Let $m$ and $m_{*}$ be Haar measure and inner Haar measure on $G$, respectively. Given $\varepsilon>0$, we classify all pairs $A,B$ of Haar measurable subsets of $G$ satisfying $m(A), m(B)>\varepsilon$ and $m_{*}(A+B)\leq m(A)+m(B)+\delta$ where $\delta=\delta(\varepsilon)>0$ is small. We also study the case where the $\delta$-popular sumset $A+_{\delta}B:=\{t\in G: m(A\cap (t-B))>\delta\}$ is small. We prove that for all $\varepsilon>0$, there is a $\delta>0$ such that if $A$ and $B$ are subsets of a compact abelian group $G$ having $m(A), m(B)>\varepsilon$ and $m(A+_{\delta}B)\leq m(A)+m(B)+\delta$, then there are sets $S, T\subseteq G$ such that $m(A\triangle S)+m(B\triangle T)<\varepsilon$ and $m(S+T)\leq m(S)+m(T)$. Appealing to known results, the latter inequality yields strong structural information on $S$ and $T$, and therefore on $A$ and $B$.