On a Sumset Conjecture of Erdős

Abstract

AbstractErdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.