Shrinking target problems for beta-dynamical system

For any β 〉 1, let （[0, 1],Tβ） be the beta dynamical system. For a positive function ψ ： N → R＋ and a real number x0 E [0, 1], we define D（Tβ, ψ, xo） the set of ψ-well approximable points by xo as {x C [0, 1] ： ]Tβ^nx - x0｜ （ψ（n） for infinitely many n ∈ N}.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 D（Tβ, ψb, x0） completely for any β 〉 1 and any positive function ψ.

For any β>1,let([0,1],Tβ) be the beta dynamical system.For a positive function ψ:N→R+ and a real number x0 ∈[0,1],we define D(Tβ,ψ,x0) the set of ψ-well approximable points by x0as {x∈[0,1]:|Tβnx-x0|<ψ(n) for infinitely many n∈N}.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 D(Tβ,ψ,x0) completely for any β>1 and any positive function ψ.