Suppose you have an infinite tower of subsets of the integers: $\displaystyle A_1 \subsetneq A_2 \subsetneq \cdots \subset \mathbb{Z}$ such that for any $\displaystyle x \in \mathbb{Z}$, there exists $\displaystyle k \in A_n$ such that $\displaystyle x \equiv k \pmod{2^n}$. Can I determine whether or not $\displaystyle x \in \bigcup_{n\ge 1}A_n$?