We prove that if is a subset of the primes, and the lower density of in the primes is larger than , then all sufficiently large odd positive integers can be written as the sum of three primes in . The constant in this statement is the best possible.
"A density version of the Vinogradov three primes theorem." Duke Math. J. 163 (3) 489 - 512, 15 February 2014. https://doi.org/10.1215/00127094-2410176