A note on Pollard's Theorem

2008/04/16 by Yahya Ould Hamidoune, Hamidoune, Y. O., Oriol Serra +1 · 1 citation
Mathematics · #Limits and Structures in Graph Theory #Advanced Topology and Set Theory #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.0804.2593

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 ∑ _{1≤ i ≤ t} |N_i (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.

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• Number of A+B≠ C solutions in abelian groups and application to counting independent sets in hypergraphs

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