Probability a random walk hits zero at specified time set

Question

Let $X_n \in \lbrace -1, 0, 1 \rbrace$ be sequence of i.i.d random variables taking $-1$ or $1$ with equal probability, and $0$ some positive probability. $S_n = \sum_{i = 1}^{n} X_i$ is a random walk. It is easy to see that $S_n$ is null recurrent. Let Q be the set of indexes so that $S_n = 0$, namely the set of return to zero times. Let $W \subset \mathbb{N}$ be any infinite subset of $\mathbb{N}$. Is it true or false that \begin{align*} Pr[W \cap Q = \emptyset] = 0 \end{align*}