Semicontinuity of structure for small sumsets in compact abelian groups

Abstract

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 $m_{\ast }$ be Haar measure and inner Haar measure on $G$, respectively. Given $\epsilon >0$, we classify all pairs $A,B$ of Haar measurable subsets of $G$ satisfying $m(A),m(B)>\epsilon$ and $m_{\ast }(A+B)\le m(A)+m(B)+\delta$ where $\delta =\delta (\epsilon )>0$ 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 $\epsilon >0$, there is a $\delta >0$ such that if $A$ and $B$ are subsets of a compact abelian group $G$ having $m(A),m(B)>\epsilon$ and $m(A{+}_{\delta }B)\le m(A)+m(B)+\delta$, then there are sets $S,T\subseteq G$ such that $m(A△S)+m(B△T)<\epsilon$ and $m(S+T)\le m(S)+m(T)$. Appealing to known results, the latter inequality yields strong structural information on $S$ and $T$, and therefore on $A$ and $B$.

56 pages. v.2 incorporates referee suggestions. v.3 applies Discrete Analysis style; minor changes to exposition