Diophantine analysis in beta-dynamical systems and Hausdorff dimensions

Let {xn}n≥1⊂[0,1] be a sequence of real numbers and let φ:N→(0,1] be a positive function. Using the mass transference principle established by Beresnevich and Velani [1], we prove that for any x∈(0,1], the Hausdorff dimension of the set{β>1:|Tβnx−xn|<φ(n) for infinitely many n∈N} satisfies a so-called 0-1 law according to limsupn→∞log⁡φ(n)n=−∞ or not, where Tβ is the β-transformation.