Density versions of the binary Goldbach problem

TL;DR

This paper proves that dense subsets of primes with relative density above 1/2 can almost always represent even numbers as sums of two primes within the subset, establishing a density version of the Goldbach problem.

Contribution

It introduces a density threshold of 1/2 for subsets of primes to almost always represent even numbers as sums of two primes within the subset, and shows this threshold is optimal.

Findings

Density > 1/2 guarantees almost all even numbers are sums of two primes in the subset.

The threshold 1/2 is proven to be the best possible.

Existence of subsets with density close to 1 that miss a positive proportion of even integers.

Abstract

Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $\delta$, then almost all even integers can be written as the sum of two primes in $A$. The constant $1/2$ in the statement is best possible. Moreover we give an example to show that for any $\varepsilon > 0$ there exists a subset of the primes with relative density at least $1 - \varepsilon$ such that $A+A$ misses a positive proportion of even integers.