Sumsets in primes containing almost all even positive integers

TL;DR

This paper proves that under certain distribution assumptions, the sumset of a prime subset contains almost all even integers, advancing understanding of additive properties of primes and their subsets.

Contribution

It establishes that well-distributed prime subsets have sumsets with density 1/2, nearly covering all even integers, improving previous results in this area.

Findings

Sumset of prime subsets has density 1/2 in natural numbers.

Almost all even integers can be expressed as sums of two primes from the subset.

Results extend classical Goldbach problem estimations.

Abstract

Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed (in the Siegel-Walfisz sense) in any arithmetic progressions to moduli $q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in the natural numbers as $x$ tends to infinity, which also yields almost all even positive integers could be represented as the sums of two primes in $A$ as $x$ tends to infinity. This result, improving the previous results in such special case, could be compared with the classical estimation for the exceptional set of binary Goldbach problem.