  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.
