Romanoff's theorem and sums of two squares

Let $A=\{a_{n}\}_{n=1}^{\infty}$ and $B=\{b_{n}\}_{n=1}^{\infty}$ be two sequences of positive integers. Under some restrictions on $A$ and $B$, we obtain a lower bound for a number of integers $n$ not exceeding $x$ that can be expressed as a sum $n = a_i + b_j$. In particular, we obtain the result in the case when $A$ is the set of numbers representable as the sum of two squares.