Shrinking target problems for beta-dynamical system

Abstract

For any β > 1, let ([0, 1], T _β ) be the beta dynamical system. For a positive function ψ: ℕ → ℝ⁺ and a real number x ₀ ∈ [0, 1], we define $\mathbb{D}\left( {T_\beta ,\psi ,x_0 } \right)$ the set of ψ-well approximable points by x ₀ as

$$\left\{ {x \in \left[ {0,1} \right]:\left| {T_\beta ^n x - x_0 } \right| < \psi \left( n \right) for infinitely many n \in \mathbb{N}} \right\}.$$

In this note, by proving a structure lemma that any ball B(x, r) contains a regular cylinder of comparable length with r, we determine the Hausdorff dimension of the set $\mathbb{D}\left( {T_\beta ,\psi ,x_0 } \right)$ completely for any β> 1 and any positive function ψ.