Thinking about the question Four polynomials representing all integers modulo m lead me to the following complementary question:

If $S$ is a set of positive integers, say that a positive integer $m$ is *covered* by $S$ if every congruence class $\bmod m$ has a representative in $S$. Denote by $C(S)$ the set of positive integers covered by $S$. If $x>0$ let $S(x) = \{ k \in S : k \le x \}$ and the *lower density* of $S$, $\ell(S) := \lim \inf_{x \rightarrow \infty} |S(x)|/x$. My question: is there a non-trivial lower bound on $\ell(C(S))$ in terms of $\ell(S)$? That is, is there a continuous function $f : [0,1] \rightarrow [0,1]$, not identically 0, such that $\ell(C(S)) \ge f(\ell(S))$.

The set in the question I referred to has density 0, so my question wouldn't apply to it. However, I wondered if there were a simple argument in the case of positive lower density. This has the smell of the kind of question that Erdos would ask, so I wouldn't be surprised to see it there.