Additive analytic number theory is a different field from arithmetic combinatorics, and it also differs from additive combinatorics. The goals and the methods are different.

The solution of the ternary Goldbach problem by Vinogradov belongs to additive analytic number theory. Vinogradov used the so-called circle method (or Hardy-Littlewood method). (Update: Anurag Sahay kindly pointed out in a comment that subsequently other methods have been found for the same problem.) Applications of this method are always technical, because one needs to carefully analyze the underlying generating series (which are trigonometric series, also known as exponential sums), e.g. how it behaves around rational points of small denominators ("major arcs") and away from such points ("minor arcs"). I recommend that you study first the solution of Waring's problem with this method. A good source is Vaughan's book "The Hardy-Littlewood method", where Chapter 2 (pp. 8-25) discusses Waring's problem and Chapter 3 (pp. 27-36) discusses Goldbach's problem. A solid background in analytic number theory helps (good textbooks are Davenport: Multiplicative number theory and Montgomery-Vaughan: Multiplicative number theory I).

As a side remark, it might appear odd that one needs to study multiplicative number theory before additive number theory. The reason is that in additive number theory we add things that are defined in terms of multiplication (e.g. powers in Waring's problem and primes in Goldbach's problem). As the famous physicist Lev Landau once said: "Why add prime numbers? Prime numbers are made to be multiplied, not added."

Additive analytic number theory is a different field from arithmetic combinatorics, and it also differs from additive combinatorics. The goals and the methods are different.

The solution of the ternary Goldbach problem by Vinogradov belongs to additive analytic number theory. Vinogradov used the so-called circle method (or Hardy-Littlewood method). (Update: Anurag Sahay kindly pointed out in a comment that subsequently other methods were found for the same problem.) Applications of this method are always technical, because one needs to carefully analyze the underlying generating series (which are trigonometric series, also known as exponential sums), e.g. how it behaves around rational points of small denominators ("major arcs") and away from such points ("minor arcs"). I recommend that you study first the solution of Waring's problem with this method. A good source is Vaughan's book "The Hardy-Littlewood method", where Chapter 2 (pp. 8-25) discusses Waring's problem and Chapter 3 (pp. 27-36) discusses Goldbach's problem. A solid background in analytic number theory helps (good textbooks are Davenport: Multiplicative number theory and Montgomery-Vaughan: Multiplicative number theory I).

As a side remark, it might appear odd that one needs to study multiplicative number theory before additive number theory. The reason is that in additive number theory we add things that are defined in terms of multiplication (e.g. powers in Waring's problem and primes in Goldbach's problem). As the famous physicist Lev Landau once said: "Why add prime numbers? Prime numbers are made to be multiplied, not added."

Additive analytic number theory is a different field from arithmetic combinatorics, and it also differs from additive combinatorics. The goals and the methods are different.

The solution of the ternary Goldbach problem by Vinogradov belongs to additive analytic number theory. The method used (which is still the only method known for the problem) is the so-called circle method (or Hardy-Littlewood method). Applications of this method are always technical, because one needs to carefully analyze the underlying generating series (which are trigonometric series, also known as exponential sums), e.g. how it behaves around rational points of small denominators ("major arcs") and away from such points ("minor arcs"). I recommend that you study first the solution of Waring's problem with this method. A good source is Vaughan's book "The Hardy-Littlewood method", where Chapter 2 (pp. 8-25) discusses Waring's problem and Chapter 3 (pp. 27-36) discusses Goldbach's problem. A solid background in analytic number theory helps (good textbooks are Davenport: Multiplicative number theory and Montgomery-Vaughan: Multiplicative number theory I).

As a side remark, it might appear odd that one needs to study multiplicative number theory before additive number theory. The reason is that in additive number theory we add things that are defined in terms of multiplication (e.g. powers in Waring's problem and primes in Goldbach's problem). As the famous physicist Lev Landau once said: "Why add prime numbers? Prime numbers are made to be multiplied, not added."

Additive analytic number theory is a different field from arithmetic combinatorics, and it also differs from additive combinatorics. The goals and the methods are different.

The solution of the ternary Goldbach problem by Vinogradov belongs to additive analytic number theory. Vinogradov used the so-called circle method (or Hardy-Littlewood method). (Update: Anurag Sahay kindly pointed out in a comment that subsequently other methods have been found for the same problem.) Applications of this method are always technical, because one needs to carefully analyze the underlying generating series (which are trigonometric series, also known as exponential sums), e.g. how it behaves around rational points of small denominators ("major arcs") and away from such points ("minor arcs"). I recommend that you study first the solution of Waring's problem with this method. A good source is Vaughan's book "The Hardy-Littlewood method", where Chapter 2 (pp. 8-25) discusses Waring's problem and Chapter 3 (pp. 27-36) discusses Goldbach's problem. A solid background in analytic number theory helps (good textbooks are Davenport: Multiplicative number theory and Montgomery-Vaughan: Multiplicative number theory I).

As a side remark, it might appear odd that one needs to study multiplicative number theory before additive number theory. The reason is that in additive number theory we add things that are defined in terms of multiplication (e.g. powers in Waring's problem and primes in Goldbach's problem). As the famous physicist Lev Landau once said: "Why add prime numbers? Prime numbers are made to be multiplied, not added."