I have $k$ distinct prime numbers $\ell_1 < \dots <\ell_k$, and for each $i=1,\dots,k$, a subset $A_i$ of $\mathbb Z / \ell_i \mathbb Z$. Let $L=\ell_1 \dots \ell_k$. Now using the chinese reminder theorem, $\mathbb Z/ L \mathbb Z = \prod_{i=1}^k \mathbb Z /\ell_i \mathbb Z$, hence the subset $\prod_{i=1}^k {A_i}$ of the RHS is identified to a subset $A$ of $\mathbb Z/L \mathbb Z$ (that what I call a "product set").

Now consider an interval $I$ of $\mathbb Z / \ell \mathbb Z$. I would say (that's acknowledgedly a vague definition) that $I$ and $A$ are "approximately independent" of $\frac{|A \cap I|}{|I|}$ is close to $\frac{|A|}{L}$. For exemple, when $I = \mathbb Z/L\mathbb Z$, then these two fractions are trivially equal.

Now it seems natural to believe that when $|I|$ is large with respect to the $\ell_i$, even if it is small with respect to $L=\prod_i \ell_i$, then $A$ is approximately independent to $I$.

Is this intuition true in some sense?

I apologize for this vague question: I think it makes sense as it is and it seems that any of my attempt to make it more precise will result in something false or trivial. I also have the frustrating impression that I have already seen this problem, or something very close to it (that is concerning the "independence" of product sets in $\mathbb Z / L \mathbb Z$ with other natural subsets of $\mathbb Z / L \mathbb Z$) discussed somewhere, perhaps even on MO. But I am not able to recall where, and missing even a name or keywords for this problem, I don't know where to look for. So any reference or names for this question is welcome. (PS: I don't even know how to tag this question. Please feel free to change tags)