Hongze Li

A density version of Vinogradov's three primes theorem

Abstract Let 𝒫 denote the set of all primes. Suppose that P 1, P 2, P 3 are three subsets of 𝒫 with d 𝒫(P 1) + d 𝒫(P 2) + d 𝒫(P 3) > 2, where d 𝒫(Pi ) is the lower density of Pi relative to 𝒫. We prove that for every sufficiently large odd integer n, there exist pi ∈ Pi such that n = p 1 + p 2 + p 3.

The Romanoff theorem revisited

Let P be the set of all primes and P_2=P\cup{p_1p_2: p_1,p_2\in P}

Monochromatic integers adding to polynomials of prime variables

. We prove that the sumset 2^P+P_2={2^p+q: p\in P, q\in P_2} has a positive lower density.

The exceptional set of Goldbach numbers

We extend a recent result of Khalfalah and Szemerédi to the polynomials of prime variables.