Suppose we have iid r.v.s $X_n \in \mathbb{R}$ with mean $0$ variance $\sigma^2$, I wonder is it true that we have $\exists a>0,$ $$P(S_n \in [-a,a] \text{ infinitely often} )=1,$$
where $S_n =\sum_{i=1}^n X_i$. In the case of integer valued $X_n$, the above is true by strong law of large numbers and Kesten-Spitzer-Whitman theorem.
By Hewitt–Savage $0-1$ law, we know the above probability is $0$ or $1$. But it looks hard to me to determine if it is positive or not.