Let $\mathcal{A}$ be the set of all arithmetric progressions in $\mathbb{N}$ i.e

\begin{align*} \mathcal{A} = \{a + b\mathbb{N} : a,b\in\mathbb{N}, b\neq 0\}. \end{align*}

Does there exist a set $X \subseteq \mathbb{N}$ such that for each $A \in \mathcal{A}$, both $A \cap X$ and $A \cap X^c$ are infinite?