Let us call a set $D\subseteq\mathbb Z$ *residue-class dense* if for each residue class $[a]_n=\{kn+a\mid k\in\mathbb Z\}$, there is a residue class $[b]_m$ with $[b]_m\subseteq [a]_n\cap D$.

Using the Sun-tzu (Chinese) Remainder Theorem we can see that examples of dense sets include the non-primes and the non-squares.
Moreover, if $D_1$ and $D_2$ are residue-class dense then so is $D_1\cap D_2$.

Elements of such dense sets can be thought of as generic integers. Thus, a generic integer is not prime, and not square.

I'm curious if this is a well-studied notion, under another name?