TITLE: How often a random walk with irrational increments is close to 0?
QUESTION [8 upvotes]: Let $\omega$ be an irrational number, and $X$ a random variable taking values $1,-1,\omega,-\omega$ each with probability $1/4$. Let then $X_i$ be iid variables with the same law as $X$ and $S_n=\sum_{i=1}^n X_i,n\in \mathbb N$ be the corresponding random walk.
Is it possible to have a precise asymptotics for $P(|S_n|<\epsilon)$ for $\epsilon>0$? Ultimately I would like to know the behaviour of 
$$\sum_{n=1}^\infty n^{-3/2} P(|S_n|<\epsilon)$$ as $\epsilon\to 0$.
I feel like the diophantine properties of $\omega$ are relevant for this asymptotics.
How would you proceed to get such an estimate? Ideally I would like to consider $X$ with any discrete law, with eventually infinitely many atoms.
EDIT: to be clear, I think there are ad-hoc methods to solve this kind of problems, as Mateusz shows below. I want to make sure not to miss any kind of general theory that solves this kind of problems in the theory of random walks.

REPLY [4 votes]: Just an extended comment. Let $X_n$ be the simple random walk in $\mathbb{Z}^2$. Then $$\mu(\{x\}) = \sum_{n = 1}^\infty n^{-3/2} P(X_n = x)$$ is comparable with $$\sum_{n = 1}^\infty n^{-3/2} \times n^{-1} \exp(-|x|^2 / (2 n)) \approx (1 + |x|)^{-3}.$$ So your question boils down to estimating $$\sum_{k \in \mathbb{Z}} \frac{1}{(1 + |k|)^3} \, \mathbb{1}_{(-\epsilon, \epsilon)}(k \omega - \lfloor k \omega\rfloor) . $$ This indeed seems closely related to how well one can approximate $\omega$ with rationals.