Zero-one law of hausdorff dimensions of the recurrent sets

Let (Σ, σ) be the one-sided shift space with m symbols and R_n(x) be the first return time of x ϵ Σ to the n-th cylinder containing x. Denote E^φ_α,β = {x ϵ Σ : lim inf n→∞ log R_n(x) φ(n) = α, lim sup n→∞ log R_n(x) φ(n) = β}, where φ : N → R⁺ is a monotonically increasing function and 0 ≤ α ≤ β ≤ +∞. We show that the Hausdorff dimension of the set E^φ_α,β admits a dichotomy: it is either zero or one depending on φ,α and β.