A note on Pollard's Theorem

Abstract

Let $A,B$ be nonempty subsets of a an abelian group $G$. Let $N_i(A,B)$ denote the set of elements of $G$ having $i$ distinct decompositions as a product of an element of $A$ and an element of $B$. We prove that $$\sum _{1\le i\le t}|N_i(A,B)|\ge t(|A|+|B|-t-\alpha +1+w)-w,$$ where $\alpha$ is the largest size of a coset contained in $AB$ and $w=\min (\alpha -1,1)$, with a strict inequality if $\alpha \ge 3$ and $t\ge 2$, or if $\alpha \ge 2$ and $t=2$. This result is a local extension of results by Pollard and Green--Ruzsa and extends also for $t>2$ a recent result of Grynkiewicz, conjectured by Dicks--Ivanov (for non necessarily abelian groups) in connection to the famous Hanna Neumann problem in Group Theory.