A note on Pollard’s Theorem

Let A,B be nonempty subsets of a an abelian group G. Let Ni(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 ∑ 1≤i≤t |Ni(A,B)| ≥ t(|A|+ |B| − t− α+ 1 + w) − w, where α is the largest size of a coset contained in AB and w = min(α − 1, 1), with a strict inequality if α ≥ 3 and t ≥ 2, or if α ≥ 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. MSC Classification: 11B60, 11B34, 20D60.