Pollard's theorem in general abelian groups

Abstract

We make further progress towards a Kneser-type generalization of Pollard's Theorem to general abelian groups. For two sets $A$ and $B$ in an abelian group $G$, the \emph{$t$-popular sumset} of $A$ and $B$, denoted by $A{+}_{t}B$, is the set of elements in $G$ each with at least $t$ representations of the form $a+b$, where $a\in A$ and $b\in B$. For $|A|,|B|\ge t\ge 2$, we prove that if \begin{align*} \sum_{i=1}^t |A+_i B|< t|A|+t|B|-\frac{4}{3}t^2+\frac{2}{3}t, \end{align*} then there exist $A^{\prime }\subseteq A$ and $B^{\prime }\subseteq B$ with $|A\setminus A^{\prime }|+|B\setminus B^{\prime }|\le t-1$, $A^{\prime }{+}_t{B}^{\prime }=A^{\prime }+B^{\prime }=A{+}_{t}B$, and ${\sum }_{i=1}^t|A{+}_{i}B|\ge t|A|+t|B|-t|H|,$ where $H$ is the stabilizer of $A^{\prime }+B^{\prime }=A{+}_{t}B$. Our result improves the main quadratic term in the previous best bound from $-2t^2$ to $-\frac{4}{3}{t}^2$.