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

We offer a complete answer to the following question on the growth of sumsets in commutative groups. Let h be a positive integer and $A,B₁,...,B_h$ be finite sets in a commutative group. We bound $|A+B₁+...+B_h|$ from above in terms of |A|, |A + B₁|, ..., $|A+B_h|$ and h. Extremal examples, which demonstrate that the bound is asymptotically sharp in all parameters, are furthermore provided.

We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

Given an additively written abelian group G and a set X ⊆ G, we let (X) denote the monoid of zero-sum sequences over X and (X) the Davenport constant of (X), namely the supremum of the positive integers n for which there exists a sequence x₁⋯xₙ in (X) such that ${\sum }_{i\in I}{x}_i\ne 0$ for each non-empty proper subset I of 1,...,n. In this paper, we mainly investigate the case when G is a power of ℤ and X is a box (i.e., a product of intervals of G). Some mixed sets (e.g., the product of a group by a box) are studied...

For an arbitrary (not totally real) number field $K$ of degree $\ge 3$, we ask how many perfect powers ${\gamma }^p$ of algebraic integers $\gamma$ in $K$ exist, such that $\mu \left(\tau \left({\gamma }^p\right)\right)\le X$ for each embedding $\tau$ of $K$ into the complex field. ($X$ a large real parameter, $p\ge 2$ a fixed integer, and $\mu \left(z\right)=max\left(\right|\mathrm{Re}\left(z\right)|,|\mathrm{Im}\left(z\right)\left|\right)$ for any complex $z$.) This quantity is evaluated asymptotically in the form $c_{p,K}{X}^{n/p}+R_{p,K}\left(X\right)$, with sharp estimates for the remainder $R_{p,K}\left(X\right)$. The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation...

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....