Density ternary Goldbach for primes in a fixed residue class

TL;DR

The paper proves a density threshold for subsets of primes in a fixed residue class to represent large multiples of 3 as sums of three primes from that subset, establishing the threshold as optimal.

Contribution

It establishes the exact density threshold of 1/2 for subsets of primes in a fixed residue class to guarantee such representations, which was previously unknown.

Findings

Density threshold of 1/2 is necessary and sufficient.

Every large multiple of 3 can be expressed as a sum of three primes from the subset.

The threshold is proven to be optimal.

Abstract

We prove that if $A$ is a subset of those primes which are congruent to $1 \pmod{3}$ such that the relative density of $A$ in this residue class is larger than $\frac{1}{2},$ then every sufficiently large odd integer $n$ which satisfies $n \equiv 0 \pmod{3}$ can be written as a sum of three primes from $A.$ Moreover the threshold of $\frac{1}{2}$ for the relative density is best possible.