Semicontinuity of structure for small sumsets in compact abelian groups

- Applied Mathematics and Statistics, Colorado School of Mines

### Editorial introduction

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|≥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)≤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, Rødseth, 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)≤m(A)+m(B)+δ when δ 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+δB to be the set of all “δ-popular” elements of A+B – that is, the set of all x∈G such that m{a∈A:x−a∈B}≥δ. The author shows that for every ϵ>0 there exists δ>0 such that if m(A+δB)≤m(A)+m(B)+δ, then there exist approximations A′ and B′ such that m(A△A′)+m(B△B′)<ϵ and m(A′+B′)≤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)≤m(A)+m(B)+δ. No explicit dependence of δ 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)≤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.