A New Proof of Vinogradov's Three Primes Theorem

TL;DR

This paper presents a novel proof of Vinogradov's three primes theorem using a transference principle and additive combinatorics, avoiding traditional reliance on L-functions.

Contribution

It introduces a new proof method based on additive combinatorics and transference principles, differing from classical L-function approaches.

Findings

Proof applies to all sufficiently large odd integers

Develops an additive combinatorial result on popular sums

Provides an alternative proof technique for prime sum theorems

Abstract

We give a new proof of Vinogradov's three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of L-functions, either explicitly or implicitly. Our proof uses instead a transference principle, the idea of which was first developed by Green. To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.