On Extending Pollard's Theorem for t-Representable Sums

Let $t\ge 1$, let $A$ and $B$ be finite, nonempty subsets of an abelian group $G$, and let $A\text{pp}iB$ 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 $A^{\mathrm{\prime }}\subseteq A$ and $B^{\mathrm{\prime }}\subseteq B$ 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 $A\text{pp}tB$ and $\rho =\mathrm{\mid }{A}^{\mathrm{\prime }}+H\mathrm{\mid }-\mathrm{\mid }{A}^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{B}^{\mathrm{\prime }}+H\mathrm{\mid }-\mathrm{\mid }{B}^{\mathrm{\prime }}\mathrm{\mid }$. In the case $t=2$, we improve (\ref{almost}) to $|A\pp{1}B|+|A\pp{2}B|\geq 2|A|+2|B|-4$.