Variants of Romanoff's theorem

Abstract

Let $\mathcal{A}=\{a_{n}\}_{n=1}^{\infty}$ and $\mathcal{B}=\{b_{n}\}_{n=1}^{\infty}$ be two sequences of positive integers (not necessarily distinct). Under some restrictions on $\mathcal{A}$ and $\mathcal{B}$, we obtain a lower bound for a number of integers $n$ not exceeding $x$ that can be represented as a sum $n = a_i + b_j$.