A generalization of Kneser's Addition Theorem

Let A=(A1,…,Am)A=(A1,…,Am) be a sequence of finite subsets from an additive abelian group G  . Let Σℓ(A)Σℓ(A) denote the set of all group elements representable as a sum of ℓ elements from distinct terms of A, and set H=stab(Σℓ(A))={g∈G:g+Σℓ(A)=Σℓ(A)}. Our main theorem is the following lower bound:|Σℓ(A)|⩾|H|(1−ℓ+∑Q∈G/Hmin{ℓ,|{i∈{1,…,m}:Ai∩Q≠∅}|}). In the special case when m=ℓ=2m=ℓ=2, this is equivalent to Kneser's Addition Theorem, and indeed we obtain a new proof of this result. The special case when every AiAi has size one is a new result concerning subsequence sums which extends some recent work of Bollobás–Leader, Hamidoune, Hamidoune–Ordaz–Ortuño, Grynkiewicz, and Gao, and resolves two recent conjectures of Gao, Thangadurai, and Zhuang.