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

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