Abstract
AbstractWe 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 is sieve theoretical and uses a transference principle, the idea of which was first developed by Green [Ann. of Math. (2) 161 (3) (2005), 1609–1636] and used in the proof of Green and Tao’s theorem [Ann. of Math. (2) 167 (2) (2008), 481–547]. To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.
Highlights
In this paper, we study additive problems involving primes
The binary Goldbach problem is considered to be beyond the scope of current techniques, its ternary analog was settled by Vinogradov [29] in 1937
There exists a positive integer V such that every odd positive integer N V can be written as the sum of three primes
Summary
We study additive problems involving primes. The famous Goldbach conjecture asserts that every even positive integer at least four is the sum of two primes. The main innovation of the current paper is a new version of Theorem 1.2, which applies even when the average possible, we will work directly in Z. of ai is Let us slightly less first state the thcoanmb21i.nTaotomriaalkreesthuilst when ai is bounded by the constant function 1. We are interested in the regime where K is a small positive constant times the cardinality of A1 or A2 In this direction, Green and Ruzsa [10] obtained the following generalization of Kneser’s theorem in arbitrary finite abelian groups. Our main result is a generalization of Theorem 2.2 to popular sums, which essentially states that the same lower bound above holds for DK ( A1, A2) when K = γ N for some small γ > 0, under some regularity assumption on A1. In the wider region K < 4, see [4] for a recent result
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