On a Sumset Conjecture of Erdős

Erdős conjectured that for any set A of natural numbers with positive lower asymptotic density, there are infinite sets B,C such that B+C is included in A. We verify Erdős’ conjecture in the case that A has Banach density exceeding 1/2. As a consequence, we prove that, for A with positive Banach density (a much weaker assumption than positive lower density), we can find infinite B,C 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.