Consider a sequence $R=(r_i\bmod m_i)_i$ of residue classes, and define $A$ as the set of positive integers $a$ such that $a=r_i\pmod m_i$ for some $i,$ and let $f(x)=|\{m_i: a_i\le x\}|$.
Erdős [1] proved that if (1) $r_i=0$ for all $i$, (2) $f(x)=O(x/\log x)$, and (3) the only $m_j$ dividing $m_i$ is $m_i$ itself, then $A$ has a density.
I am interested in weakening condition (1), that is, considering general residue classes rather than only multiples. Are any results known along these lines? I'm prepared to weaken (2) to $O(\sqrt x)$ if needed.
[1] P. Erdős, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948), pp. 685–692.