The ternary Goldbach problem with primes in positive density sets

TL;DR

This paper proves that for large odd integers, sums of three primes from sets with certain density thresholds can represent all sufficiently large odd numbers, extending the classical Goldbach problem to positive density subsets.

Contribution

It establishes the minimal density conditions under which three primes from specified sets can sum to all large odd integers, generalizing the Goldbach problem.

Findings

If the lower density of P1 exceeds 5/8, and P2, P3 have densities at least 5/8, then every large odd integer can be expressed as their sum.

The density condition of 5/8 is shown to be optimal for this representation.

The result extends Goldbach's conjecture to subsets of primes with positive density.

Abstract

Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if $\underline{\delta}(P_{1})>5/8$, $\underline{\delta}(P_{2})\geq5/8$, and $\underline{\delta}(P_{3})\geq5/8$, then for every sufficiently large odd integer n, there exist $p_{i} \in P_{i}$ such that $n=p_{1}+p_{2}+p_{3}$. The condition is the best possible.