Pollard's theorem in general abelian groups
David J. Grynkiewicz, Runze Wang
TL;DR
This work advances a Kneser-type generalization of Pollard's Theorem to general abelian groups by analyzing the $t$-popular sumset $A+_t B$ and proving a structural result under a refined bound. The authors establish that if $\sum_{i=1}^t |A+_i B|$ falls below a specific threshold, there exist near-identity removals $A'\subseteq A$, $B'\subseteq B$ with $A'+_t B'=A'+B'=A+_t B$ and a lower bound controlled by the stabilizer $H=\mathsf H(A'+B')$, thereby improving the quadratic term from $-2t^2$ to $-\frac{4}{3}t^2$. The main result strengthens prior partial results (e.g., Hamidoune–Serra, Grynkiewicz) and yields a fully explicit conclusion for $t=2$, while outlining the limit of current techniques for larger $t$. The paper also demonstrates an application to restricted sumsets, obtaining new lower bounds for $|A\overset{\tau}{+} B|$ in terms of the group order and the sizes of $A$ and $B$, thus contributing to the broader program of unifying Pollard-type and Kneser-type phenomena in 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 $|A|,\, |B|\ge t\geq 2$, 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 $A'\subseteq A$ and $B'\subseteq B$ with $|A\setminus A'|+|B\setminus B'|\le t-1$, $A'+_t B'=A'+B'=A+_t B$, and $ \sum_{i=1}^t |A+_i B|\ge t|A|+t|B|-t|H|,$ where $H$ is the stabilizer of $A'+B'=A+_t B$. Our result improves the main quadratic term in the previous best bound from $-2t^2$ to $-\frac{4}{3}t^2$.