Let $t\geq 1$, let $A$ and $B$ be finite, nonempty subsets of an abelian group $G$, and let $A\pp{ i} B$ denote all the elements $c$ with at least $i$ representations of the form $c=a+b$, with $a\in A$ and $b\in B$. For $|A|, |B|\geq t$, we show that either \be\label{ almost}\Sum{ i=1}{ t}|A\pp{ i} B|\geq t|A|+t|B|-2t^2+1,\ee or else there exist $A'\subseteq A$ and $B'\subseteq B$ with \ber \nn l&:=&|A\setminus A'|+|B\setminus B'|\leq t-1, \nn A'\pp{ t}B'&=&A'+B'=A\pp{ t}B,{ and} \nn \Sum{ i=1}{ t}|A\pp{ i}B|&\geq& t|A|+t|B|-(t-l)(|H|-\rho)-tl\geq t|A|+t|B|-t|H|,\eer where $H$ is the (nontrivial) stabilizer of $A\pp{ t} B$ and $\rho=|A'+H|-|A'|+|B'+H|-|B'|$. In the case $t=2$, we improve (\ref{ almost}) to $|A\pp{ 1}B|+|A\pp{ 2}B|\geq 2|A|+2|B|-4$.
Categories: math.NT math.CO