A sieve by not (necessarily) coprime integers

Question

Let $\mathcal{A}$ be a nonempty set of pairwise coprime positive integers, and for each $a \in \mathcal{A}$ let $\Omega_a \subsetneq \{0, \ldots, a - 1\}$ be a set of residues modulo $a$. Furthermore, put $$\mathcal{S} := \{n \in \mathbf{N} : (n \bmod a) \notin \Omega_a, \;\forall a \in \mathcal{A}\}.$$ An easy application of the Chinese Remainder Theorem shows that $$\underline{\mathbf{d}}(\mathcal{S}) \geq \prod_{a \in \mathcal{A}} \left(1 - \frac{\#\Omega_a}{a}\right),$$ where $\underline{\mathbf{d}}$ denotes the lower asymptotic density. In particular, if $$\sum_{a \in \mathcal{A}} \frac{\#\Omega_a}{a} < +\infty$$ then the infinite product converges and $\mathcal{S}$ has positive asymptotic density. (Actually, the asymptotic density of $\mathcal{S}$ exists and it is equal to the infinite product, but this is irrelevant for my question.)

My question is: What if we drop the hypothesis that the elements of $\mathcal{A}$ are pairwise coprime? Are there results that guarantee that $\mathcal{S}$ has a positive lower asymptotic density? Have these kind of sieves been studied?

Of course, if the elements of $\mathcal{A}$ are not pairwise coprime then (without additional hypotheses) $\underline{\mathbf{d}}(\mathcal{S})$ can be $0$, just consider: $\mathcal{A} = \{2, 4\}$, $\Omega_2 = \{0\}$, $\Omega_4 = \{1, 3\}$.

Thank you for any suggestion.

Answer 1

This paper by I Ruzsa (On the small sieve. II. Sifting by composite numbers. J. Number Theory 14 (1982), no. 2, 260–268.) gives considerable limitations to the type of lower bound that you want: If $H(x,K)$ is the maximum number of integers $n \leq x$, not divisible by integers $a \in A$. Here the maximum is taken over sets of integers $A$, not containing $1$ and with $\sum_i \frac{1}{a_i} \leq K$. Then $H(x,K)$ can be quite small. Let $K>1$, then $H(x,K)$ is about $x^{e^{1-K}}$, more precisely: $\lim_{x \rightarrow \infty} \log H(x,K)/\log x=e^{1-K}$. Moreover, for $K=1$ one has: $\frac{c_1 x}{\log x} < H(x,1)< \frac{x}{(\log x)^{c_2}}$.

In other words, already the case of $\Omega_a=\{0\}$ makes problems. If you still want a lower density bound you need to avoid the case above...