On a sumset conjecture of Erd\H{o}s

TL;DR

This paper proves Erdős's sumset conjecture for sets with Banach density over 1/2 and extends results to countable amenable groups, also addressing pseudorandom sets.

Contribution

It verifies Erdős's conjecture for sets with Banach density > 1/2 and generalizes the results to amenable groups and pseudorandom sets.

Findings

Confirmed Erdős's conjecture for sets with Banach density > 1/2.

Extended sumset results to countable amenable groups.

Provided solutions for pseudorandom subsets of natural numbers.

Abstract

Erd\H{o}s conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erd\H{o}s' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ 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 Erd\H{o}s' conjecture for subsets of the natural numbers that are pseudorandom.