Density ternary Goldbach for primes in a fixed residue class

Abstract

We prove that if $A$ is a subset of those primes which are congruent to $1(\mathrm{mod}3)$ 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 $n\equiv 0(\mathrm{mod}3)$ 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.

14 pages