A result on the approximation properties of the orbit of 1 under the β-transformation

For any real number β>1, we denote by Tβ the β-transformation on the unit interval [0,1] given by Tβ(x)=βx−⌊βx⌋, where ⌊ξ⌋ denotes the integer part of ξ. We consider the size of the set of β>1 where the orbit of 1 under Tβ ultimately has a positive distance from a given point in [0,1]. For any x0∈[0,1] and any (β0,β1)⊂(1,∞), we obtain the set of β∈(β0,β1) such that x0 is not an accumulation point of the orbit of 1 under Tβ with full Hausdorff dimension. Specifically,dimH{β∈(β0,β1):liminfn→∞|Tβn1−x0|>0}=1. This is a generalization of the result of Schmeling for the case where x0=0 and (β0,β1)=(1,∞).