Semicontinuity of structure for small sumsets in compact abelian groups

Abstract

Semicontinuity of structure for small sumsets in compact abelian groups, Discrete Analysis 2019:18, 46 pp. The well-known Cauchy-Davenport theorem asserts that if $A$ and $B$ are two subsets of a cyclic group of prime order $p$, then $|A+B|\geq\min\{|A|+|B|-1,p\}$. It was generalized to a suitable statement about finite subsets of arbitrary Abelian groups by Martin Kneser: note that if $A$ and $B$ are unions of cosets of a subgroup $H$, then $|A+B|$ can be as small as $|A|+|B|-|H|$, and Kneser's theorem takes account of this. The question of what one can say when the inequalities are sharp was answered by Kemperman, who provided a rather complicated structural characterization. One can ask a corresponding question when $G$ is a compact Hausdorff Abelian topological group with Haar measure $m$. Now we let $A$ and $B$ be $m$-measurable subsets such that $$m_*(A+B)\leq m(A)+m(B).$$ Here $m_*$ is the inner $m$-measure, since $A+B$ does not have to be measurable. (Indeed, Sierpinski showed that there are two measure-zero sets $A,B$ of reals such that $A+B$ is not measurable.) Such pairs of sets were characterized by Kneser under the additional assumption that $G$ is connected. An obvious example is where $A$ and $B$ are subintervals of the circle group, and Kneser showed that, roughly speaking, every example is an inverse image of such an example under a surjective $m$-measurable homomorphism. When $G$ is disconnected the characterization of pairs satisfying $m_*(A+B)=m(A)+m(B)$ is more complicated. Building on work of Hamidoune, Rodseth, Serra, and Zemor, Grynkiewicz provided a complete characterization of such pairs for discrete abelian groups $G$. The author of this paper combined Grynkiewicz's and Kneser's proofs to extend this to arbitrary compact Hausdorff abelian groups. The aim of this paper is to prove a stability version of preceding results: this is the meaning of the phrase "semicontinuity of structure" in the title. In other words, the paper is concerned with what happens if $m_*(A+B)\leq m(A)+m(B)+\delta$ when $\delta$ is sufficiently small as a function of $m(A)$ and $m(B)$. One of the main results is the following, which has a similar flavour to the triangle removal lemma. Define $A+_\delta B$ to be the set of all "$\delta$-popular" elements of $A+B$ -- that is, the set of all $x\in G$ such that $m\{a\in A: x-a\in B\}\geq\delta$. The author shows that for every $\epsilon>0$ there exists $\delta>0$ such that if $m(A+_\delta B)\leq m(A)+m(B)+\delta$, then there exist approximations $A'$ and $B'$ such that $m(A\triangle A')+m(B\triangle B')<\epsilon$ and $m(A'+B')\leq m(A')+m(B')$. Since, as was mentioned above, pairs $A',B'$ with this stronger property have been classified, this gives a complete characterization of pairs $A,B$ with the weaker property. The proof of this result leads to the desired classification of pairs $A,B$ for which $m_*(A+B)\leq m(A)+m(B)+\delta$. No explicit dependence of $\delta$ on $m(A)$ and $m(B)$ is provided, because ultraproduct methods are used in the proof. These results had already been proved by Tao for connected abelian groups $G$, but the generalization to disconnected groups is not straightforward. Whereas Tao's proof recovers Kneser's characterization of sets $A,B$ with $m_*(A+B)\leq m(A)+m(B)$ for connected abelian groups, the present work uses the detailed structure of such sets, in the absence of connectedness, as a building block.