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

A New Proof of Vinogradov's Three Primes Theorem