Zero-one law of Hausdorff dimensions of the recurrent sets

Let $(\mathrm{\Sigma },\sigma )$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\in \mathrm{\Sigma }$ to the $n$-th cylinder containing $x$. Denote $$E_{\alpha ,\beta }^{\phi }=\{x\in \mathrm{\Sigma }:\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\log R_n(x)}{\phi (n)}=\alpha ,\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\log R_n(x)}{\phi (n)}=\beta \},$$ where $\phi :\mathbb{N}\to {\mathbb{R}}^{+}$ is a monotonically increasing function and $0\le \alpha \le \beta \le +\mathrm{\infty }$. We show that the Hausdorff dimension of the set $E_{\alpha ,\beta }^{\phi }$ admits a dichotomy: it is either zero or one depending on $\phi ,\alpha$ and $\beta$.