The Ternary Goldbach Problem In The Spinit Of Green-Tao

The ternary Goldbach conjecture states that every odd integer greater than 7 is the sum of three primes. This problem was basically solved by Vinogradov [12] in 1937, and in fact he showed that for every sufficiently large odd integer n,whereand A > 0 is arbitrary. Nowadays this Goldbach-Vinogradov theorem has become a classical result in additive number theory. Later, using a similar method, van der Corput [2] proved that the primes contain infinitely many non-trivial 3-term arithmetic progressions.On the other hand, another classical result due to Roth [9] asserts that a set A of integers contains infinitely many non-trivial 3-term arithmetic progressions provided that (?)(A) > 0, whereRoth's theorem is a special case of the well-known Szemeredi theorem [18], which states that any set A of integers with (?)(A) > 0 contains arbitrarily long arithmetic progressions. For a set X of positive integers and its subset A, define the upper density of A relative to X byLet P denote the set of all primes. In [4], Green obtained a Roth-type generalization of van der Corput's result. Green showed that if P₀ is a subset of P with (?)_P(P₀) > 0 then P₀ contains infinitely many non-trivial 3-term arithmetic progressions . One major ingredient in Green's proof is a transference principle, which transfers a subset of primes with relative positive density to a subset of Z_N = Z/NZ (where N is a large prime) with positive density. Subsequently, this principle was greatly improved (in a different way) in the proof of Green and Tao's celebrated theorem [5] that the primes contains arbitrarily long arithmetic progressions. The Hardy-Littlewood circle method [11] is commonly applied in Vinogradov's, van der Corput's, Roth's and Green's proofs. Recently, using Green's idea, Li Hongze and Pan Hao [8] extended the Goldbach-Vinogradov theorem.In this thesis, we, inspired by Green and Tao's transference principle in [5], will give another proof of Li and Pan's theorem. In Chapter 1, we will introduce Li and Pan's theorem. Chapter 2 is the most important part of this thesis, in which we shall use Green-Tao's idea to show a so-called transference principle. Using the transference principle in Chapter 2, we give a new proof of Li and Pan's theorem in Chapter 3.