## A density version of the Vinogradov three primes theorem.(English)Zbl 1330.11062

The famous ternary Goldbach problem states that every odd integer greater than 7 is the sum of three odd primes. A classical theorem of I. M. Vinogradov [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 15, 169–172 (1937; Zbl 0016.29101)] shows that every sufficiently large odd integer is the sum of three primes.
In the paper under review, the author applies methods from additive combinatorics to prove that if $$A$$ is a subset of the primes whose lower density exceeds $$5/8$$, then every sufficiently large odd integer is the sum of three primes in $$A$$. Under some suitable local conditions, it is shown that the lower bound of $$5/8$$ can be replaced by $$1/2$$. This work is motivated by the work of H. Li and H. Pan [Forum Math. 22, No. 4, 699–714 (2010; Zbl 1232.11103)], in which the three primes are taken from three possibly different subsets, the sum of whose lower densities exceeds 2. This obviously implies a version of the author’s result with 5/8 replaced by 2/3.
A Cauchy-Davenport-Chowla-type result on the corresponding sumsets modulo odd integers $$m$$ plays a key role in the proof of the main result of the paper under review.

### MSC:

 11P32 Goldbach-type theorems; other additive questions involving primes 11D85 Representation problems

### Citations:

Zbl 1232.11103; Zbl 0016.29101
Full Text:

### References:

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