A generalization of Kneser's Addition Theorem

Let A = (A(1),..., A(m)) be a sequence of finite subsets from an additive abelian group G. Let Sigma(l) (A) denote the set of all group elements representable as a sun, of l elements from distinct terms of A, and set H = stab(Sigma(l)(A)) = {g is an element of G: g + Sigma(l)(A) = Sigma(l)(A)}. Our main theorem is the following lower bound: vertical bar Sigma(l)(A)vertical bar >=vertical bar H vertical bar(1 - l + Sigma(Q is an element of G/H) min {l,vertical bar{i is an element of {1,....m}: A(i) boolean AND Q not equal empty set}vertical bar}). In the special case when m = 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 A(i) has size one is a new result concerning subsequence sums which extends some recent work of Bollobas-Leader, Hamidoune, Hamidoune-Ordaz-Ortuno, Grynkiewicz. and Gao, and resolves two recent conjectures of Gao, Thangadurai, and Zhuang. (C) 2008 Elsevier Inc. All rights reserved.