Let t ≥ 1, let A and B be finite, nonempty subsets of an abelian group G, and let $nAmathop + limits_i Bn$nAmathop + limits_i Bn denote all the elements c with at least i representations of the form c = a + b, with a ∈ A and b ∈ B. For |A|, |B| ≥ t, we show that either $nsumlimits_{i = 1}^t {|Amathop + limits_i B| geqslant t|A| + t|B| - 2t^2 + 1,} n$nsumlimits_{i = 1}^t {|Amathop + limits_i B| geqslant t|A| + t|B| - 2t^2 + 1,}
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机译：令t≥1，令A和B为阿贝尔群G的有限的非空子集，令$ nAmathop + limits_i Bn $ nAmathop + limits_iBn表示所有元素c，至少i形式为c = a + b ，其中a∈A和b∈B。对于| A |，| B | ≥t，我们表明$ nsumlimits_ {i = 1} ^ t {| Amathop + limits_i B | geqslant t | A | + t | B | -2t ^ 2 + 1，} n $ nsumlimits_ {i = 1} ^ t {| Amathop + limits_i B | geqslant t | A | + t | B | -2t ^ 2 + 1，}
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