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**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.

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.

Reviewer: Zhixin Liu (Tianjin)

### MSC:

11P32 | Goldbach-type theorems; other additive questions involving primes |

11D85 | Representation problems |

### References:

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