Semicontinuity of structure for small sumsets in compact abelian groups

John T. Griesmer

Discrete analysis (2019)

Résumé

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 $m_{*}(A+B)\leq m(A)+m(B)+δ$ where $δ=δ(\varepsilon)>0$ is small. We also study the case where the $δ$-popular sumset $A+_δB:=\{t\in G: m(A\cap (t-B))>δ\}$ is small. We prove that for all $\varepsilon>0$, there is a $δ>0$ such that if $A$ and $B$ are subsets of a compact abelian group $G$ having $m(A), m(B)>\varepsilon$ and $m(A+_δB)\leq m(A)+m(B)+δ$, 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 $B$.

Détail

Publié le : 2019-11-29