A short proof of Erd\H{o}s's $B+C$ conjecture

TL;DR

This paper provides a concise proof that sets of natural numbers with positive upper Banach density necessarily contain the sum of two infinite subsets, simplifying previous proofs.

Contribution

It introduces a shorter, more straightforward proof of Erdős's B+C conjecture for sets with positive upper Banach density.

Findings

Sets with positive upper Banach density contain the sum of two infinite sets.

The proof simplifies earlier approaches to the conjecture.

The result confirms a key property of dense subsets of natural numbers.

Abstract

We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.