A generalization of Kneser's Addition Theorem

Abstract Let A = ( A 1 , … , A m ) be a sequence of finite subsets from an additive abelian group G . Let Σ l ( A ) denote the set of all group elements representable as a sum of l elements from distinct terms of A , and set H = stab ( Σ l ( A ) ) = { g ∈ G : g + Σ l ( A ) = Σ l ( A ) } . Our main theorem is the following lower bound: | Σ l ( A ) | ⩾ | H | ( 1 − l + ∑ Q ∈ G / H min { l , | { i ∈ { 1 , … , m } : A i ∩ Q ≠ ∅ } | } ) . 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.

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