Hi everybody.

In "M. B. Nathanson - Elementary Methods in Number Theory" is shown (Theorem 7.14) that if $A$ is a set of positive integers such that $\sum_{a \in A} 1 / a$ converges then the set of multiples of $A$ has a natural density (a set of positive integers $S$ has a natural density if exists $\lim_{x \to \infty} |S \cap [1,x]| / x$).
Equivalently, the set of positive integers $n$ such that $a \nmid n$ for all $a \in A$ has a natural density. 

I'm looking for a generalization of this kind:

Let $M$ be a set of positive integers and $N_m \subseteq \mathbb{Z} / m \mathbb{Z}$ for all $m \in M$. If [some conditions on $|N_m| / m$, maybe that $\sum_{m \in M} |N_m| / m$ converges] then the set of positive integers $n$ such that $n \not \equiv r \mod m$ for all $m \in M$ and $r \in N_m$ has a natural density.

Thank you for any reference :-)