Consider $k\geq 2$ biinfinite arithmetic progressions $\mathcal A_i=a_i+b_i\mathbb Z$ (for $i=1,\ldots,k$) in $\mathbb Z$. (We suppose that $a_i$ and $b_i\geq 2$ are strictly positive integers. One can assume $a_i$ and $b_i$ coprime and $a_i$ can be reduced modulo $b_i$ without loss of generality.)

The set $\mathcal P=\prod_{i=1}^k \mathcal A_i$ of all possible products has strictly positive upper and lower densities $\overline{\delta}$ and $\underline{\delta}$ in $\mathbb Z$.

*Is it possible that $\overline{\delta}>\underline{\delta}$?
Can they be irrational? Are they easy to compute? Fixing $\epsilon>0$ and $k\geq 2$ are there only finitely many such products $\mathcal P$ with upper density at least $\epsilon$?*

Examples: $(1+2\mathbb Z)(1+3\mathbb Z)=\mathbb Z\setminus\{0\}$ (and both densities are $1$).

$(1+2\mathbb Z)(1+4\mathbb Z)=1+2\mathbb Z$ (and both densities are equal to $1/2$).

$(1+3\mathbb Z)(1+5\mathbb Z)$ seems to have densities above $.75$.