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 ( mod 3 ) such that the relative density of A in this residue class is larger than 1 2 , then every sufficiently large odd integer n which satisfies n ≡ 0 ( mod 3 ) can be written as a sum of three primes from A . Moreover the threshold of 1 2 for the relative density is best possible.