A Density Version of the Vinogradov Three Primes Theorem

TL;DR

This paper proves that any sufficiently large odd number can be expressed as the sum of three primes from a subset of primes with density greater than 5/8, establishing a density threshold for such representations.

Contribution

It introduces a density version of the Vinogradov three primes theorem, identifying the optimal density threshold for subsets of primes.

Findings

Density > 5/8 guarantees representations of large odd integers as three primes from the subset.

The 5/8 constant is proven to be optimal.

The result extends classical additive prime number theory to dense subsets.

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.