If $A$ is an infinite subset of $\mathbb N$, a random subset $X\subseteq\mathbb N$ will satisfy the condition $|A\cap X|=|A\cap X^c|=\aleph_0$ with probability one. Inasmuch as there are only countably many arithmetic progressions, a random subset will satisfy that condition for all of them with probability one.

If $A$ is an infinite subset of $\mathbb N$, a random subset $X\subseteq\mathbb N$ will satisfy the condition $|A\cap X|=|A\cap X^c|=\aleph_0$ with probability one. Inasmuch as there are only countably many arithmetic progressions, a random subset will satisfy that condition for all of them with probability one.

Alternatively, just define $X$ to be the set of all natural numbers with an odd number of digits.