Pollard's theorem in general abelian groups

TL;DR

This paper extends Pollard's Theorem to general abelian groups by establishing bounds on the sum of sizes of popular sumsets, improving previous quadratic bounds and identifying structural properties of the sets involved.

Contribution

It provides a generalized Kneser-type inequality for popular sumsets in abelian groups, with improved bounds on the sum of their sizes and structural characterizations.

Findings

Established a new bound on the sum of popular sumset sizes in abelian groups.

Proved the existence of subsets with specific sumset properties close to the original sets.

Improved the quadratic term in the sumset size bound from -2t^2 to -4/3 t^2.

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 $-\frac{4}{3}t^2$.