A density version of the Vinogradov three primes theorem

Abstract

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible.

An -Function-Free Proof of Vinogradov’s Three Primes Theorem

“…This directly follows by Green's argument (see [19,27]). It is usually applied as follows in studying additive problems involving dense subsets of primes.…”

“…https://doi.org/10.1017/fms.2014. 27 We are now ready to prove Theorem 4.1. Let R = N 1−δ/2 and Q = N δ/4 be parameters.…”

“…https://doi.org/10.1017/fms.2014. 27 If the functions a i all have the same average, then the mean condition above is satisfied when α i > 2 5 , beating the 1 2 barrier. Theorem 1.4 will be deduced from a robust version of Freiman's 3k − 3 theorem in Section 2.…”

The ternary Goldbach problem with primes in positive density sets

“…then for every sufficiently large odd integer n, there exist p i ∈ P i (i = 1, 2, 3) such that n = p 1 + p 2 + p 3 . Motivated by the work of Li and Pan, X. Shao proved [5] (2014) that if A is a subset of P with δ(A) > 5 8 , then for every sufficiently large odd integer n, there exist p i ∈ A (i = 1, 2, 3) such that n = p 1 + p 2 + p 3 . It is worth mentioning that X. Shao gave [6] (2014) an l-function-free proof of Vinogradov's three primes theorem.…”

Sums of positive density subsets of the primes

“…This is sharp in the sense that there are examples in which α 1 + α 2 + α 3 = 2 but the claim is not true. However X. Shao [15] has very recently shown that in case P 1 = P 2 = P 3 it is enough that α i > 5/8 (which is again sharp).…”