We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1-o\left(1\right))\alpha /\left(e^{\gamma }loglog(1/\beta )\right)$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $\mathbb{Z}{*}_m$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A,B\subseteq \mathbb{Z}{*}_m$ of densities α and β, the...

We extend two results of Ruzsa and Vu on the additive complements of primes.

Suppose that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ are nonzero real numbers, not all negative, $\delta >0$, $𝒱$ is a well-spaced set, and the ratio ${\lambda }_1/{\lambda }_2$ is algebraic and irrational. Denote by $E(𝒱,N,\delta )$ the number of $v\in 𝒱$ with $v\le N$ such that the inequality $$|{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^3+{\lambda }_3{p}_3^4+{\lambda }_4{p}_4^5-v|<v^{-\delta }$$ has no solution in primes $p_1$, $p_2$, $p_3$, $p_4$. We show that $$E(𝒱,N,\delta )\ll N^{1+2\delta -1/72+\epsilon }$$ for any $\epsilon >0$.

1. Introduction. A positive number which is a sum of two odd primes is called a Goldbach number. Let E(x) denote the number of even numbers not exceeding x which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that E(x) = 2 for every x ≥ 4. E(x) is usually called the exceptional set of Goldbach numbers. In [8] H. L. Montgomery and R. C. Vaughan proved that $E\left(x\right)=O\left(x^{1-\Delta }\right)$ for some positive constant Δ > 0$.In\left[3\right]ChenandPanprovedthatonecantake\Delta >0.01.In\left[6\right],weprovedthatE\left(x\right)=O\left(x^{0.921}\right)$. In this paper we prove the following result. Theorem....